MA1101 — Mathematics for Electronics IMathematics for Electronics I

Why This Matters

An electronic product is specified by quantities: supply voltage, current, resistance, frequency, power, gain, timing error, and noise level. Numbers alone are not enough. Engineering requires relationships between quantities, together with units and assumptions. Mathematics gives those relationships a form that can be checked and used.

1.1 Quantities, numbers, and units

A physical quantity is a measurable property together with a unit. For example, \(5\) is only a number; \(5\ \mathrm{V}\) is a voltage. The same care is needed when writing equations: both sides of a physically meaningful equation must have the same units.

Real numbers are the main number system of this course. They include natural numbers such as \(0,1,2,\ldots\), integers such as \(-2,-1,0,1,2\), rational numbers that can be written as \(p/q\) with \(q\ne0\), and irrational numbers such as \(\sqrt2\) and \(\pi\). Visualise the real numbers as points on a number line: values increase as we move to the right, decrease as we move to the left, and there is no largest or smallest real number.

Scientific notation keeps very large and very small quantities readable:

The prefixes kilo-, milli-, micro-, and nano- are shorthand for powers of ten. Always estimate the order of magnitude before trusting a numerical output. A calculated capacitor voltage of \(3.3\times10^9\ \mathrm{V}\) in a \(5\ \mathrm{V}\) laboratory circuit is usually a model or unit error, not an exciting discovery.

1.2 Variables and equations

A variable is a symbol representing a quantity that may vary or whose value is currently unknown. An equation asserts that two expressions have the same value. In the resistor model

\(V\) is voltage, \(I\) is current, and \(R\) is resistance. If \(R\) is constant and \(V\) changes, then \(I\) changes as well. The equation makes that relationship testable.

A disciplined modelling workflow is:

1. State the situation and the assumptions.
2. Name each relevant quantity and include units.
3. Write the basic relationships.
4. Solve only for quantities that the model actually determines.
5. Check the result using units, signs, magnitude, and limiting cases.

1.3 Expressions and algebraic equivalence

An expression is a meaningful combination of numbers, variables, and operations, such as \(3x^2-4x+1\). Two expressions are equivalent if they have the same value for every input where both are defined. For example,

is true only for \(x\ne3\). The left-hand expression is undefined at \(x=3\), while the right-hand expression is defined there. Algebraic cancellation simplifies an expression; it does not silently repair a forbidden input.

1.4 Geometry as a model-building tool

Equations often encode geometry. A rectangle of base \(b\) and perpendicular height \(h\) has area \(A=bh\). A parallelogram with the same base and perpendicular height has the same area: cut a triangular piece from one side and move it to the other side to form a rectangle. This is why its area is also \(bh\).

The word perpendicular matters. A slanted side of a parallelogram is not automatically its height. The area of a triangle with base \(b\) and perpendicular height \(h\) is half the matching parallelogram area:

This kind of rearrangement argument is a useful habit: replace a difficult shape or equation by an equivalent one whose meaning is easier to see.

Why This Matters

When a quantity depends on another quantity, a graph lets us see the relationship before we calculate with it. Voltage may depend on time, current may depend on voltage, and a measured output may depend on a control setting. Functions make such dependencies precise.

2.1 Cartesian coordinates

The Cartesian plane has a horizontal \(x\)-axis and a vertical \(y\)-axis. A point \((x,y)\) means move \(x\) units horizontally from the origin and then \(y\) units vertically. The order matters: \((2,5)\) and \((5,2)\) are different points.

To visualise a graph of \(y=f(x)\), imagine choosing an input \(x\), calculating the associated output, and plotting the point \((x,f(x))\). Repeating this process creates a curve or collection of points.

2.2 What a function is

A function assigns exactly one output to each allowed input. We write

where \(D\) is the domain of permitted inputs and \(R\) is a set containing the outputs. The range is the set of outputs actually attained.

For \(f(x)=x^2\), if the domain is all real numbers, then the range is \([0,\infty)\). The value \(f(-3)=9\) is a number. The function \(f\) is the whole rule, not the particular output \(9\).

The vertical-line test gives a graphical version of the definition: if a vertical line meets a graph more than once, some input has two outputs, so the graph does not represent a single-valued function of \(x\).

2.3 Function algebra

Given functions \(f\) and \(g\), we can define

The quotient is defined only where \(g(x)\ne0\). Function algebra is useful for building models from simpler pieces. For example, a sinusoidal input plus a constant offset has the form \(v(t)=V_0+A\sin(\omega t)\).

2.4 Graphs as solution tools

To solve an equation \(f(x)=g(x)\) graphically, plot \(y=f(x)\) and \(y=g(x)\) on the same axes. Their intersection points have equal \(y\)-coordinates, so their \(x\)-coordinates solve the equation. A graph provides insight and estimates; it does not replace an exact algebraic method when close roots or scale distortions matter.

2.5 Transformations and interpretation

Starting from \(y=f(x)\):

• \(y=f(x)+k\) shifts the graph upward by \(k\).
• \(y=f(x-h)\) shifts it right by \(h\).
• \(y=af(x)\) scales output values vertically.
• \(y=f(-x)\) reflects the graph in the \(y\)-axis.

These are not rules to memorise without meaning. For instance, replacing \(x\) by \(x-h\) asks the original function to receive the value \(x-h\). To produce the old feature at input \(0\), the new graph needs \(x-h=0\), hence \(x=h\): the feature moves right.

2.6 Domain, range, and transformations as graph-reading tools

The domain answers “which inputs make sense?”; the range answers “which outputs can the rule actually produce?” To find a domain, exclude zero denominators, negative even-root inputs, and invalid logarithms. To find a range, use a graph or rewrite the expression to expose a lowest, highest, or excluded value.

2.7 Reading a graph as a mathematical statement

A graph simultaneously communicates domain, range, zeros, intercepts, increasing and decreasing portions, symmetry, and limiting behaviour. Read its axes before reading a numerical value: a plotted “3” may mean \(3\) V, \(3\) ms, or \(3\times 10^{-3}\) depending on the scale. Algebraic form often predicts the graph before any points are plotted.

Why This Matters

Linear relationships describe proportional change and local approximations. Quadratic relationships describe curvature, extrema, and roots. Together they form the algebraic vocabulary needed for circuit equations, optimisation, and later calculus.

3.1 Straight lines and slope

A non-vertical straight line has equation

The constant \(m\) is the slope: change in \(y\) per unit change in \(x\). The constant \(b\) is the \(y\)-intercept, because \(x=0\) gives \(y=b\).

For two distinct points \((x_1,y_1)\) and \((x_2,y_2)\) on a non-vertical line,

Why does this work? The line equation gives \(y_1=mx_1+b\) and \(y_2=mx_2+b\). Subtracting eliminates \(b\):

Dividing by the nonzero horizontal change gives the slope formula. A vertical line has \(x_2-x_1=0\), so its slope is undefined and its equation is \(x=c\).

3.2 Point-slope and two-point forms

Starting with \(y=mx+b\) and a point \((x_1,y_1)\) on the line,

This point-slope form is often the quickest way to write a line. Substituting the two-point slope gives

provided the line is not vertical and \(x\ne x_1\) during the displayed division. The undivided point-slope form remains safer in symbolic work.

3.3 Intersections and solution cases

The intersection of \(y=m_1x+b_1\) and \(y=m_2x+b_2\) satisfies both equations. Equating them gives

If \(m_1\ne m_2\), then

and there is one intersection. If \(m_1=m_2\) but \(b_1\ne b_2\), the lines are parallel and there is no solution. If both slope and intercept agree, the two equations describe the same line and there are infinitely many solutions.

3.4 Quadratic functions and their shape

A quadratic function has the form

Its graph is a parabola. The sign of \(a\) controls opening: \(a>0\) gives a bowl opening upward; \(a<0\) gives a bowl opening downward. To see the centre and extreme value clearly, rewrite the quadratic in vertex form.

Start by factoring \(a\) from the first two terms:

Inside the bracket, add and subtract \(\left(\frac{b}{2a}\right)^2\):

The first three terms form a square, hence

The vertex is therefore

3.5 Roots and the quadratic formula

Roots solve \(ax^2+bx+c=0\). Using the completed-square expression,

From the completed-square form, move the constant term carefully:

Multiply by \(4a\):

Taking square roots and solving for \(x\) gives

The quantity \(\Delta=b^2-4ac\) is the discriminant. If \(\Delta>0\), there are two distinct real roots; if \(\Delta=0\), one repeated real root; and if \(\Delta<0\), no real roots. Complex roots are studied in Chapter 14.

3.6 Lines and parabolas as geometric models

Slope is a rate with units. If voltage is plotted against time, slope has units V/s; an intercept has voltage units. A negative slope says the output decreases as input increases. A line through two measurements is an interpolation model, not proof that the same rule remains valid far outside the measurements.

Problem-Solving Strategy

For a line, choose slope-intercept form when the slope/intercept are known and point-slope form when a point and slope are known. For a quadratic, use vertex form to study shape and extrema, factorisation when factors are visible, and the quadratic formula when a general reliable method is needed.

Why This Matters

One equation describes one relationship. A circuit node, a network of constraints, or a polynomial fitted to data usually produces several relationships at once. A linear system lets us solve them together and, just as importantly, tells us when the data are inconsistent or leave freedom in the answer.

4.1 Linear systems as models

A linear equation in variables \(x_1,\ldots,x_n\) has the form

The variables occur only to the first power and are not multiplied by each other. A linear system consists of several such equations.

4.2 Elimination and back-substitution

Start
[ 1  1 | 3 ]
[ 2 −1 | 0 ]

R₂ ← R₂ − 2R₁
[ 1  1 | 3 ]
[ 0 −3 | −6 ]

R₂ ← −⅓R₂
[ 1  1 | 3 ]
[ 0  1 | 2 ]

R₁ ← R₁ − R₂
[ 1  0 | 1 ]
[ 0  1 | 2 ]

Consider

The \(y\)-coefficients are already opposites, so add the two equations directly:

The \(y\) terms cancel, leaving \(3x=9\), hence \(x=3\). Substituting into the second equation gives \(3-y=2\), so \(y=1\). The cancellation was deliberate: we chose an operation that removes one unknown while preserving the solution set.

The guiding idea is to replace equations by equivalent equations that remove one variable. Valid row operations are:

1. Exchange two equations.
2. Multiply one equation by a nonzero constant.
3. Add a multiple of one equation to another.

They preserve the solution set because each new equation is a logical consequence of the old system and the operation can be reversed.

4.3 Augmented matrices

The coefficients and right-hand sides can be recorded compactly as an augmented matrix:

The vertical bar separates the coefficient matrix from the constants. Matrix notation reduces bookkeeping; it does not change the underlying equations.

4.4 Row-echelon form, pivots, and rank

After elimination, a system is easiest to read in row-echelon form: each successive nonzero row begins farther to the right. The first nonzero entry in a nonzero row is a pivot. The number of pivots in the coefficient matrix is its rank, denoted \(r\).

For a consistent system with \(n\) unknowns:

• \(r=n\) means every variable is determined: there is one solution.
• \(r<n\) means \(n-r\) variables are free: there are infinitely many solutions.

An augmented row of the form

means \(0=b\), a contradiction. The system is inconsistent and has no solution.

4.5 Matrix-vector form

For

we write

Thus \(A\mathbf x=\mathbf b\). Matrix multiplication means that \(A\mathbf x\) is a linear combination of the columns of \(A\). Solving the system asks whether \(\mathbf b\) can be made from those columns, and if so, in how many ways.

4.6 Elimination is reversible bookkeeping

Replacing equation 2 by equation 2 minus three times equation 1 preserves every solution: any solution of the old equations satisfies the new equation, and adding three times equation 1 back reverses the operation. Scaling is reversible only by a nonzero number, which is why multiplying a row by zero is not allowed.

Start
[ 1  1 | 3 ]
[ 2 −1 | 0 ]

R₂ ← R₂ − 2R₁
[ 1  1 | 3 ]
[ 0 −3 | −6 ]

R₂ ← −⅓R₂
[ 1  1 | 3 ]
[ 0  1 | 2 ]

R₁ ← R₁ − R₂
[ 1  0 | 1 ]
[ 0  1 | 2 ]

Why This Matters

Many engineering ideas are iterative: a sampled signal is a sequence, a numerical algorithm produces successive approximations, and a transient circuit approaches a steady state. Limits give a precise language for “approaches.”

5.1 Sequences

A sequence is a function whose inputs are positive integers. We write

For example, \(a_n=1/n\) gives \(1,1/2,1/3,\ldots\). Unlike a continuous graph, visualise a sequence as isolated dots above the integer positions \(1,2,3,\ldots\).

A sequence may be explicit, such as \(a_n=3n-1\), or recursive, such as

The recursive rule above is Newton’s iteration for \(\sqrt2\).

5.2 Limits of sequences

We say \(a_n\) converges to \(L\), written \(a_n\to L\), if its terms can be made as close to \(L\) as desired by going sufficiently far along the sequence. Formally, for every \(\varepsilon>0\) there is an integer \(N\) such that

The value \(L\) need not appear as a sequence term. For \(a_n=1/n\), no term is zero, but terms eventually fit inside every band \((-\varepsilon,\varepsilon)\) around zero.

5.3 Boundedness and monotonicity

A sequence is bounded above by \(M\) if \(a_n\le M\) for all \(n\), and bounded below by \(m\) if \(a_n\ge m\) for all \(n\). It is non-decreasing if \(a_{n+1}\ge a_n\) and non-increasing if \(a_{n+1}\le a_n\).

The theorem guarantees existence; it does not by itself find the value of the limit. For a recursive sequence, first prove the appropriate bounds and monotonicity, then take limits on both sides of the recursion.

5.4 Series and partial sums

Given a sequence \((a_n)\), the expression

is a series. Its meaning is the limit of partial sums

Do not confuse \(a_n\to0\) with convergence of the series. It is necessary for a series to converge, but it is not sufficient: the harmonic series \(\sum_{n=1}^{\infty}1/n\) diverges even though \(1/n\to0\).

5.5 Sequence limit laws

If \(a_n\to A\) and \(b_n\to B\), then sums, products, and scalar multiples behave as expected:

For rational expressions in \(n\), divide numerator and denominator by the highest power of \(n\) that occurs.

Some sequence limits are not rational expressions. A standard one is

It says that repeatedly taking a root pulls every fixed positive number toward \(1\). One way to see the structure is to write \(a^{1/n}=e^{(\ln a)/n}\): the exponent tends to \(0\), and continuity of the exponential gives the result. This is not a substitution of infinity into an exponent; it is a composition of two ordinary limits.

5.6 Tails, oscillation, and partial sums

Convergence concerns the tail of a sequence: changing finitely many early terms does not change its limit. Picture a horizontal \(\varepsilon\)-band around \(L\). Once every later dot lies in that band, the definition is met. Oscillation alone does not prevent convergence; what matters is whether the distance to one value shrinks.

5.7 A sequence is not the series built from it

The sequence \(a_n\) is a list of individual contributions. The series \(\sum a_n\) is a new object whose values are the partial sums \(S_N=a_1+\cdots+a_N\). A convergent sequence can therefore produce either a convergent or divergent series. The term test only provides a one-way decision: if \(a_n\) fails to approach zero, the series diverges; if it does approach zero, another test is required.

Why This Matters

Calculus relies on understanding what happens near a point, not merely at it. Limits formalise this nearby behaviour. Continuity tells us when small input changes produce small output changes. Exponentials provide the model for growth, decay, and later circuit transients.

6.1 Real powers and exponential functions

For integer exponents, \(a^n\) means repeated multiplication. For arbitrary real exponents, the clean real-valued theory takes \(a>0\). Rational powers use roots, for example \(a^{m/n}=\sqrt[n]{a^m}\) when this is real and defined. Irrational exponents are defined consistently through limits of rational approximations.

The functions \(x^a\) and \(a^x\) are different. In \(x^a\), the base varies; in \(a^x\), the exponent varies. For \(a>1\), \(a^x\) passes through \((0,1)\), increases strictly, tends to \(0\) as \(x\to-\infty\), and grows without bound as \(x\to\infty\). For \(0<a<1\), the direction is reversed.

6.2 Limits of functions

We write

when \(f(x)\) becomes arbitrarily close to \(L\) as \(x\) becomes sufficiently close to \(c\), without requiring \(x=c\). The value \(f(c)\) may be different from \(L\) or may not even be defined.

The left and right limits must agree for a two-sided limit to exist:

6.3 Continuity

A function is continuous at \(c\) when all three conditions hold:

1. \(f(c)\) is defined.
2. \(\lim_{x\to c}f(x)\) exists.
3. The limit equals \(f(c)\).

Polynomials are continuous on all real numbers. Rational functions are continuous wherever their denominators are nonzero. Root functions require their real-domain restrictions. These facts are useful because they allow direct substitution only after the domain has been checked.

6.4 One-sided limits and discontinuities

A switch can change a circuit model abruptly. At the switching instant, a piecewise function may have a left limit and a different right limit. This is a jump discontinuity. A rational function may have an infinite discontinuity or vertical asymptote where its denominator approaches zero. A missing point with a finite surrounding limit is a removable discontinuity.

6.5 Limits at infinity

Limits at infinity describe long-run behaviour. For example,

The leading powers control the result. Exponentials eventually dominate any fixed polynomial power: for every fixed nonnegative integer \(m\),

This comparison later explains why exponential transients decay despite polynomial factors that may appear in intermediate calculations.

Problem-Solving Strategy

Before evaluating a limit, check the domain and try direct substitution. If it produces an indeterminate form such as \(0/0\), simplify algebraically by factorisation, rationalisation, or dominant-term division. Use one-sided limits for piecewise definitions and denominator sign changes. Do not perform algebra on the symbol \(\infty\) as though it were a real number.

6.6 Limits describe nearby behaviour, not a point calculation

Direct substitution is a useful first test, not the definition of a limit. A \(0/0\) result says numerator and denominator both become small; it does not say the quotient is zero. Factorisation or rationalisation can expose a simpler nearby expression, provided the excluded input is still recorded.

Why This Matters

An average speed over ten seconds does not tell us the speed at one instant. In the same way, the ratio \(\Delta V/\Delta t\) describes an average voltage change over an interval, while a circuit or control problem often needs the instantaneous rate of change. The derivative is the limiting version of that average rate.

7.1 Average rate of change and secant slope

For a function \(y=f(x)\), the average change in output between \(x\) and \(x+h\) is

On a graph, the two points \((x,f(x))\) and \((x+h,f(x+h))\) determine a secant line. Its slope is precisely the displayed quotient. In a physical model where \(x=t\) is time, it is average rate of change over a time interval of length \(h\).

As \(h\) becomes smaller, the second point moves towards the first. If the secant slopes approach one definite number, that number is the slope of the tangent line and the instantaneous rate of change.

7.2 Definition of derivative

The derivative of \(f\) at \(x\) is

provided the limit exists. It is a new function when it can be computed at many inputs. Other notations include \(\frac{df}{dx}\) and, when \(y=f(x)\), \(\frac{dy}{dx}\).

7.3 When a derivative does not exist

Differentiability is stronger than continuity. A differentiable function is continuous, but a continuous function need not be differentiable.

For \(f(x)=|x|\), the left-hand slopes at \(0\) are \(-1\) and the right-hand slopes are \(1\). Because these do not agree, \(f'(0)\) does not exist. A jump discontinuity also prevents differentiability. A graph may additionally have a vertical tangent, where the ordinary finite derivative fails to exist.

7.4 Linearity rules

The derivative respects addition and constant scaling:

These follow from the corresponding limit laws. The derivative of a constant is \(0\), because its output does not change, and \(\frac d{dx}x=1\).

7.5 Product and quotient rules

For a product, both factors may change. Add and subtract the intermediate term \(f(x+h)g(x)\):

Divide by \(h\) and take the limit. Continuity of differentiable functions lets \(f(x+h)\to f(x)\), yielding

For \(g(x)\ne0\), write \(f/g=f\,g^{-1}\) and differentiate, or derive directly:

7.6 Chain rule

Suppose \(y=f(u)\) and \(u=g(x)\). A small change in \(x\) first changes \(u\), which then changes \(y\). The derivative is

This is the chain rule. It is a statement about composition, not a trick about parentheses.

7.7 Power, polynomial, and rational functions

For a real exponent \(a\), the power rule

is used on intervals where the function is real and differentiable. For arbitrary real \(a\), the standard safe domain is \(x>0\). Integer powers have their usual larger domains; selected rational powers may have additional real inputs.

Term-by-term differentiation gives, for example,

For rational functions, use the quotient rule or first simplify carefully while retaining all domain restrictions.

7.8 Higher derivatives

The second derivative is the derivative of the derivative:

In motion it measures acceleration; in a graph it records how the slope changes. For a voltage signal, \(v'(t)\) and \(v''(t)\) describe increasingly rapid aspects of variation. Higher derivatives later provide Taylor approximations and determine the form of second-order differential equations.

7.9 Derivative rules as controlled local changes

The product rule has two terms because both factors can change. The chain rule has an inside derivative because the outer function receives a changing input. Before differentiating, identify the outermost operation and the independent variable; this prevents applying rules mechanically.

Why This Matters

The derivative becomes most useful when it reveals global behaviour: whether a function rises or falls, where a design reaches a maximum, and how a difference between two values is controlled by a slope in between. These conclusions require theorems with hypotheses, not pictures alone.

8.1 Rolle’s theorem and the Mean Value Theorem

If a smooth graph begins and ends at the same height, it must have a horizontal tangent somewhere in between. The endpoint and differentiability assumptions are essential; a sharp corner can defeat the conclusion.

The average slope of a secant line is therefore achieved by a tangent somewhere inside the interval. The MVT is the rigorous bridge from local derivative signs to interval-wide trends.

8.2 Monotonicity

If \(f'(x)>0\) throughout an interval, then \(f\) is strictly increasing there; if \(f'(x)<0\), it is strictly decreasing. By the MVT,

for some point \(c\) between \(x\) and \(y\). When \(y>x\), the sign of the difference is therefore the sign of \(f'(c)\).

8.3 Extrema and optimisation

An interior local maximum or minimum of a differentiable function must occur at a point where \(f'(c)=0\). Such points are critical points, but they are only candidates. A local extremum can also occur where the derivative does not exist, and an absolute extremum on a closed interval may occur at an endpoint.

8.4 L’Hôpital’s rule

The rule is not a licence to differentiate any expression in a limit. First verify the quotient form and its hypotheses. Products, differences, and powers must be transformed before the rule can be considered.

8.5 From derivative signs to reliable conclusions

Critical points are candidates, not automatic extrema. Make a sign chart by splitting the domain at derivative zeros and points where the derivative fails to exist; then test the sign on each interval. For absolute extrema on a closed interval, compare function values at every valid critical point and both endpoints.

8.6 Theorems give conclusions only under their hypotheses

The Mean Value Theorem is a guarantee about a smooth unbroken graph on a whole interval. It does not say every secant slope occurs as a tangent slope if the graph jumps, has a corner, or is undefined at an endpoint. Derivative signs use the same hypothesis through the MVT: a positive derivative on an interval gives increase only when the function is differentiable throughout its interior.

Why This Matters

Engineers repeatedly reverse relationships: infer temperature from a sensor voltage, infer resistance from current and voltage, or infer a phase from a sinusoidal measurement. Local linear approximations then tell us how errors travel through those inversions and how a complicated function can be approximated near an operating point.

9.1 Inverse functions

A function \(f:D\to R\) has an inverse on its range when it is one-to-one: no two inputs in \(D\) have the same output. Its inverse reverses the mapping:

The notation \(f^{-1}(x)\) means inverse function, not reciprocal \(1/f(x)\). A horizontal-line test expresses one-to-one behaviour graphically. The graph of an inverse is the reflection of the original graph in the line \(y=x\).

For \(f(x)=x^2\), all-real input is not one-to-one because \(f(2)=f(-2)\). Restricting to \(x\ge0\) gives an inverse \(f^{-1}(y)=\sqrt y\) for \(y\ge0\).

9.2 Derivative of an inverse

Suppose \(y=f^{-1}(x)\), so \(f(y)=x\). Differentiate with respect to \(x\):

Therefore

provided the required derivatives exist and the denominator is nonzero. An inverse may exist at a point where its derivative is not finite; the nonzero condition must not be omitted.

9.3 Linearisation and differentials

Near \(x=a\), a differentiable function is approximated by its tangent line:

Writing \(dx\) for a small input change and \(dy\) for the corresponding linearised output change gives

The differential \(dy\) is an approximation to the actual finite change \(\Delta y=f(a+\Delta x)-f(a)\). The approximation improves as \(\Delta x\) becomes smaller, assuming the local curvature is controlled.

9.4 Taylor polynomials

The best polynomial approximation near \(x=a\) matches the value and successive derivatives of \(f\) at \(a\):

\(T_1\) is the tangent-line approximation. Higher terms capture curvature and successive changes of curvature.

The theorem gives both an approximation and an error mechanism. A Taylor series is not automatically equal to the function merely because its terms can be written down; equality requires a remainder that tends to zero.

9.5 Inverse slopes, differentials, and approximation error

At corresponding points on inverse graphs, slopes are reciprocals: a steep forward graph has a shallow inverse graph. This is why the denominator in the inverse-derivative formula must be nonzero.

9.6 Taylor polynomials: matching local behaviour term by term

At \(x=a\), \(T_0\) matches only the function value, \(T_1\) also matches the tangent slope, and \(T_2\) also matches curvature. The factorials ensure that differentiating a Taylor term the required number of times leaves the matching derivative coefficient. This is why Taylor polynomials fit a function near the expansion point more faithfully as order rises.

Why This Matters

An infinite expression is meaningful only when its partial results settle to a definite value. Series make it possible to define and approximate functions that finite polynomials cannot express exactly, including the exponential function that governs capacitor transients.

10.1 Geometric series

The finite geometric sum is

Multiply by \(r\) and subtract:

Therefore, for \(r\ne1\),

If \(|r|<1\), then \(r^{N+1}\to0\) and

If \(|r|\ge1\), the terms do not tend to zero in the required way, so the series diverges. The condition is part of the formula.

10.2 Positive-term comparison and \(p\)-series

For nonnegative terms, comparison turns a difficult series into a known one. If

eventually and \(\sum b_n\) converges, then \(\sum a_n\) converges. If \(a_n\ge b_n\ge0\) eventually and \(\sum b_n\) diverges, then \(\sum a_n\) diverges.

The \(p\)-series gives a central benchmark:

The harmonic series is the \(p=1\) case. Its terms approach zero but its partial sums do not settle, demonstrating again that term limit zero is not sufficient.

10.3 Ratio and root tests

For a series \(\sum a_n\), the ratio test examines

If \(L<1\), the series converges absolutely; if \(L>1\) or \(L=\infty\), it diverges; if \(L=1\), the test says nothing. The root test has the corresponding quantity

It has the same conclusions. Factorials often make ratios simple; powers raised to the \(n\)th power often make roots simple.

10.4 Power series and radius of convergence

A power series centred at \(0\) has form

There is a radius \(R\in[0,\infty]\) such that the series converges absolutely for \(|x|<R\) and diverges for \(|x|>R\). At \(|x|=R\), each endpoint must be tested separately. When the relevant root limit exists,

For a series centred at \(a\), replace \(x\) by \(x-a\) throughout.

10.5 Functions defined by power series

Within its radius of convergence, a power series is continuous and may be differentiated term by term. The derivative has the same radius of convergence:

This theorem is powerful precisely because it is restricted to the open interval of convergence. It is not an invitation to differentiate a divergent expression.

10.6 Partial sums, test selection, and endpoint discipline

A series converges only when its partial sums settle. The term test is a fast necessary-condition check: if \(a_n\) does not tend to zero, \(\sum a_n\) diverges immediately. Passing this check proves nothing further. Choose comparison for positive rational-looking terms, ratio for factorials or exponentials, and root for an expression naturally raised to the \(n\)th power.

10.7 Choosing a convergence test and representing functions

Tests are tools with limited conclusions. A ratio or root limit equal to \(1\) is not a near-convergence result; it is no result at all from that test. For a power series, first find the open radius interval, then test each endpoint in the original series. Endpoint behaviour can differ because signs or known benchmark series change there.

Why This Matters

The equation \(y'=y\) describes a quantity whose rate of change is proportional to its current value. It appears in ideal growth and decay models and, after scaling, in first-order circuit responses. The exponential function is the function that solves it. The logarithm reverses exponential relationships.

11.1 Defining \(\exp(x)\)

The exponential function can be defined by its power series

The ratio test shows convergence for every real \(x\). Differentiating term by term returns the same series:

Let \(e=\exp(1)\). The addition law

implies \(\exp(x)=e^x\) for real \(x\). It also gives \(e^{-x}=1/e^x\), so \(e^x>0\) for all real \(x\).

11.2 Exponential growth and decay

The function \(Ae^{kt}\) has initial value \(A\). If \(k>0\), it grows; if \(k<0\), it decays. Its derivative is

The time constant \(\tau>0\) is often used in decay as \(e^{-t/\tau}\). After one time constant the remaining fraction is \(e^{-1}\approx0.368\); after five it is about \(0.0067\). These are model-based landmarks, not abrupt physical cutoffs.

11.3 Logarithms as inverses

Because \(e^x\) is strictly increasing and positive, it has an inverse \(\ln x\) on \(x>0\):

The addition law for exponentials becomes logarithm laws:

when the real expressions are defined. Differentiating \(e^{\ln x}=x\) gives

so \(\frac d{dx}\ln x=1/x\) for \(x>0\).

For \(u(x)\ne0\) on an interval,

11.4 General bases

For \(a>0\), \(a\ne1\),

This gives

11.5 Exponentials, logarithms, and model checks

The equation \(e^{u}=e^{v}\) permits \(u=v\) because the exponential is one-to-one. Taking \(\ln\) of an equation is valid only when both sides are positive real numbers. A logarithm can turn a product into a sum and an exponential unknown into a linear unknown, but it cannot repair a negative or zero argument.

Why This Matters

Derivatives describe local rate. Integration reverses the question: if a rate is known, how much has accumulated? Current is rate of charge flow, so accumulated charge is an integral of current. Area is another familiar accumulation model.

12.1 Riemann sums and signed area

For a function \(f\) on \([a,b]\), split the interval into narrow pieces of width \(\Delta x\). A rectangle of height \(f(x_k^*)\) has signed area \(f(x_k^*)\Delta x\). Adding the rectangles and taking the limit defines

Area above the axis contributes positively and area below contributes negatively. The integral is therefore signed accumulation, not automatically total geometric area.

12.2 Antiderivatives and the FTC

An antiderivative of \(f\) is a function \(F\) with \(F'=f\). Antiderivatives differ by a constant because constants differentiate to zero.

The theorem makes exact integration practical. It also says that the accumulation function \(A(x)=\int_a^x f(t)\,dt\) has derivative \(A'(x)=f(x)\) when \(f\) is continuous.

12.3 Basic antiderivatives

For \(a\ne-1\),

The excluded exponent has its own rule:

on intervals not crossing zero. Also,

12.4 Substitution

Substitution reverses the chain rule. If \(u=g(x)\) and \(du=g'(x)\,dx\), then

For a definite integral, either change the limits to \(u\)-limits or return to \(x\) before substituting the original limits.

12.5 Building a definite integral from rectangles

The symbol \(\int_a^b f(x)\,dx\) is compact, but it represents a very concrete limiting measurement. Suppose \(f(x)=x^2\) on \([0,1]\). Divide the interval into \(n\) equal pieces. Each has width \(\Delta x=1/n\). If right endpoints are used, the \(k\)th height is \((k/n)^2\), so the right-rectangle total is

The finite sum formula \(\sum_{k=1}^n k^2=n(n+1)(2n+1)/6\) gives

As \(n\to\infty\), the lower-order terms disappear and \(S_n\to1/3\). Thus

This calculation is not a replacement for the Fundamental Theorem of Calculus; it explains what that theorem evaluates efficiently. A graph should be imagined here: the curve \(y=x^2\) rises from \((0,0)\) to \((1,1)\), while the narrow rectangles leave ever smaller gaps above or below the curve.

\(n=4\), then a second panel with many thin rectangles. Label \(\Delta x\) and a sample height \(f(x_k^*)\).]

12.6 Why the Fundamental Theorem works

There are two complementary parts to the theorem. The first says that an accumulation function has the original rate as its derivative. Define

The letter \(t\) is a dummy variable: changing it to \(u\) does not change the meaning. For a small change \(h\),

The right-hand side is the average value of \(f\) over a short interval. When \(f\) is continuous, every value in that short interval becomes close to \(f(x)\) as \(h\to0\). Therefore \(A'(x)=f(x)\). This is the accumulation analogue of the secant line becoming a tangent line.

The second part says that if \(F'=f\), then the total accumulation from \(a\) to \(b\) is \(F(b)-F(a)\). Since \(A'=F'\), the functions \(A\) and \(F\) differ by a constant. Because \(A(a)=0\), that constant is \(-F(a)\), and hence

The continuity hypothesis gives a dependable ordinary Riemann integral and makes the average-value argument valid. More advanced mathematics extends the theorem further, but this course uses the stated continuous-function setting.

12.7 Substitution as a change of scale

The chain rule says

Substitution reads this equation backwards. The inner expression \(g(x)\) is a new coordinate, and \(g'(x)\,dx\) tells us how a small change in \(x\) changes that coordinate. It is not enough for an integrand to contain something that looks like an inner expression: the accompanying differential must match, possibly after extracting a constant.

12.8 A reliable integral-solving routine

Before calculating, identify the quantity being accumulated and attach units. Then inspect the integrand: a sum invites term-by-term integration; a composite expression with its inner derivative invites substitution; a product may need integration by parts in Chapter 13. For a definite integral, sketch signs, zeros, and any discontinuities. Finally differentiate an indefinite answer or check an exact definite result by a size estimate.

Why This Matters

Once an integral represents an accumulation, the remaining challenge is choosing a method that exposes an antiderivative or a geometry. Substitution handles a hidden chain rule; integration by parts handles a product. Applications require careful intervals and diagrams, not formula substitution alone.

13.1 Integration by parts

The product rule says \((uv)'=u'v+uv'\). Rearranging and integrating gives

Choose \(u\) so differentiation simplifies it, and choose \(dv\) so integration is available.

13.2 Area between curves

If \(f(x)\ge g(x)\) on \([a,b]\), area between the curves is

If the curves cross, first solve \(f(x)=g(x)\) and split the interval at each intersection. Otherwise positive and negative pieces cancel and yield signed, not physical, area.

13.3 Arc length and volumes

For a differentiable curve \(y=f(x)\), approximate a small arc by a straight segment:

Thus

When a region is rotated about an axis, thin disks or washers produce volume. For rotation about the \(x\)-axis with outer radius \(R(x)\) and inner radius \(r(x)\),

Draw the region and a representative slice before choosing bounds.

13.4 Improper integrals

An integral over an infinite interval or with an unbounded integrand is defined by a limit. For example,

It converges precisely when \(p>1\). The infinity symbol is a limiting process, not a permitted endpoint at which an antiderivative is directly evaluated.

13.5 Choosing integration by parts deliberately

Integration by parts transfers a derivative from one factor to the other. It is most useful when that transfer makes the remaining integral simpler. A practical preference order for choosing \(u\) is often: logarithmic functions, inverse trigonometric functions, algebraic polynomials, trigonometric functions, then exponentials. This is a guide, not a theorem. The decisive question is whether \(\int v\,du\) is easier than the original integral.

13.6 From a drawing to an area or volume integral

An application integral begins with geometry, not an antiderivative. Mark all intersections, choose vertical or horizontal slices, label the slice dimensions, and only then write the integral. For area with vertical slices, the height is “top minus bottom.” For a washer about the \(x\)-axis, the cross-sectional area is \(\pi(R^2-r^2)\), not \(\pi(R-r)^2\).

slice labelled top \(x\), bottom \(x^2\), height \(x-x^2\). A companion washer diagram labels outer radius \(R(x)\) and inner radius \(r(x)\).]

13.7 Arc length: local straightness, global curvature

Over a very short horizontal displacement \(dx\), a smooth curve changes vertically by approximately \(dy=f'(x)dx\). Pythagoras gives the short length

Adding these short lengths and taking a limit gives the arc-length formula from Section 13.3. The formula is not “height plus width”: a curved path is the hypotenuse of many infinitesimal right triangles.

13.8 Improper integrals and convergence checks

When an endpoint is infinite, replace it by a finite variable before integrating. For \(p\ne1\),

If \(p>1\), then \(1-p<0\) and \(b^{1-p}\to0\), so the limiting integral is \(1/(p-1)\). If \(p<1\), the power grows without bound. At \(p=1\) the integral is \(\ln b\), which also grows without bound. This establishes the threshold \(p>1\) rather than merely memorising it.

Why This Matters

The equation \(x^2+1=0\) has no real solution, yet such equations arise naturally when we analyse oscillation. Complex numbers enlarge the number system in a controlled way. Their geometry also provides the cleanest language for amplitude and phase in sinusoidal steady-state analysis.

14.1 The complex number system

Define \(i\) by \(i^2=-1\). A complex number is

where \(x\) and \(y\) are real. The real and imaginary parts are \(\operatorname{Re}z=x\) and \(\operatorname{Im}z=y\). Two complex numbers are equal only when both corresponding parts are equal.

Addition and multiplication follow algebra, with \(i^2=-1\):

The conjugate of \(z=x+iy\) is \(\bar z=x-iy\). Multiplying conjugates removes the imaginary cross terms:

This motivates the modulus

For \(z\ne0\),

14.2 Complex plane, arguments, and radians

Plot \(z=x+iy\) as the point \((x,y)\) in a plane: horizontal coordinate is real part, vertical coordinate is imaginary part. Its distance from the origin is \(|z|\). An angle from the positive real axis to the point is an argument.

Arguments are not unique: if \(\theta\) is an argument, so is \(\theta+2\pi k\) for every integer \(k\). A chosen principal argument \(\operatorname{Arg}z\) uses one stated interval, commonly \((-\pi,\pi]\).

Radians are defined by arc length divided by radius. On a unit circle, an angle \(\theta\) has arc length \(\theta\). This is why derivative and power-series formulas for sine and cosine use radians rather than degrees.

14.3 Polar coordinates: converting points and checking quadrants

The same geometry describes an ordinary Cartesian point \((x,y)\) and a complex number \(z=x+iy\). Its polar coordinates are a radius \(r\ge0\) and an angle \(\theta\) measured from the positive real axis:

For \(r>0\), the two equations determine the quadrant as well as the reference angle. A calculator command based only on \(\arctan(y/x)\) can be misleading, because the same ratio occurs in opposite quadrants and because it is undefined when \(x=0\). Use a quadrant-aware angle function such as \(\operatorname{atan2}(y,x)\) when available, or use the signs of both \(x\) and \(y\) to select the correct argument.

14.4 Polar form and multiplication

Every nonzero complex number can be written as

Later Euler’s formula writes this more compactly as \(z=re^{i\theta}\). Multiplying polar forms multiplies magnitudes and adds arguments:

Geometrically, multiplication performs a scale and a rotation. Division divides magnitudes and subtracts angles.

14.5 Arithmetic, conjugates, and geometric checks

Complex arithmetic is ordinary polynomial algebra with the one replacement \(i^2=-1\). Keeping real and imaginary parts collected is important: a complex number is zero only if both parts are zero. The conjugate changes the sign of the imaginary coordinate, so it is a reflection in the real axis. Its product with the original number is a squared distance:

That fact explains division rather than supplying a trick to memorise. A complex denominator is made real by multiplying numerator and denominator by its conjugate; this multiplies the fraction by \(1\) and changes no value.

14.6 Polar form is scale plus rotation

In rectangular coordinates, multiplication requires four products. In polar coordinates, a number has length \(r\) and direction \(\theta\). Multiplying by \(r_2e^{i\theta_2}\) scales every distance by \(r_2\) and rotates every direction by \(\theta_2\). This gives the rules

For division, \(z_2\) must be nonzero:

14.7 Roots and the multivalued argument

To solve \(w^n=z=re^{i\theta}\) with \(r>0\), the argument of \(z\) may be \(\theta+2\pi k\) for any integer \(k\). Dividing by \(n\) gives the \(n\) distinct roots

Why This Matters

Sinusoids model periodic signals. Their amplitude, frequency, phase, and rate of change all have geometric meaning on the unit circle. Complex exponentials turn trigonometric identities and phase shifts into ordinary algebra.

15.1 Sine and cosine from the unit circle

On the unit circle, the point at radian angle \(\theta\) has coordinates

Therefore

Both functions are \(2\pi\)-periodic. Sine is odd, \(\sin(-\theta)=-\sin\theta\); cosine is even, \(\cos(-\theta)=\cos\theta\).

The signal

has offset \(V_0\), amplitude \(|A|\), angular frequency \(\omega\) radians per second, period \(T=2\pi/\omega\) when \(\omega>0\), and phase \(\phi\). A phase shift moves a wave horizontally; it is not a change in amplitude.

15.2 Tangent, reciprocal functions, and related rates

The remaining elementary trigonometric functions are ratios or reciprocals of sine and cosine:

Each formula carries a restriction. For example, \(\tan x\) and \(\sec x\) are undefined where \(\cos x=0\), namely at \(x=\pi/2+k\pi\); \(\csc x\) and \(\cot x\) are undefined where \(\sin x=0\), namely at \(x=k\pi\). Tangent and cotangent have period \(\pi\), whereas secant and cosecant have period \(2\pi\). Since \(|\sin x|\le1\) and \(|\cos x|\le1\), any defined value of \(\sec x\) or \(\csc x\) has magnitude at least \(1\).

The quotient rule gives

The reciprocal rule similarly gives

These formulas are valid only at inputs where the relevant function is defined. Their reverse forms supply useful integrals on any interval that does not cross a singularity:

15.3 Derivatives and series

With radian measure,

Repeated differentiation cycles every four derivatives. Taylor’s theorem then gives

These series converge for every real \(x\). In particular, \(\lim_{x\to0}\sin x/x=1\), a central small-angle fact.

The expression \(\operatorname{sinc}x=\sin x/x\) is therefore assigned the continuous value \(\operatorname{sinc}(0)=1\). This normalized sinc function is common in signal processing; its value at zero comes from a limit, not division by zero.

15.4 Euler’s formula

Insert \(ix\) into the power series for \(e^x\). Even powers of \(i\) alternate between \(1\) and \(-1\); odd powers provide a factor \(i\). Grouping real and imaginary terms gives

This is Euler’s formula. It does not replace real sine and cosine with “imaginary tricks”; it packages their linked behaviour into one expression.

Adding angles now follows from multiplication:

Use Euler’s formula on both sides:

Multiplication gives a real part and an imaginary part,

Equality of complex numbers requires equality of the corresponding parts, so

Putting \(b=-\pi/2\) shows \(\cos(x-\pi/2)=\sin x\): sine and cosine are the same shape with a quarter-cycle phase shift. This derivation is often more reliable than memorising several separate identities.

15.5 Roots of unity

The \(n\)th roots of unity solve \(z^n=1\). In polar form, their distinct values are

They are equally spaced around the unit circle. Complex conjugate roots occur in pairs when polynomial coefficients are real; paired factors can combine into real quadratic factors. For example, roots \(e^{i\theta}\) and \(e^{-i\theta}\) produce

which has real coefficients. The imaginary components cancel because the roots are conjugates.

15.6 Reading and constructing a sinusoid

A sinusoid is best understood as a moving unit-circle coordinate. In \(A\cos(\omega t+\phi)\), the input angle advances at \(\omega\) radians each second. One complete turn requires \(2\pi\) radians, which explains

The amplitude is \(|A|\), not necessarily \(A\): a negative coefficient can be absorbed as a phase shift because \(-\cos\theta=\cos(\theta+\pi)\). The horizontal shift is found by setting the internal angle to a landmark value. For example, \(\cos(\omega t+\phi)\) has a maximum when \(\omega t+\phi=2\pi k\).

15.7 Identities as geometry and algebra

The Pythagorean identity comes from the unit-circle equation \(x^2+y^2=1\). Dividing it by \(\cos^2x\), where \(\cos x\ne0\), gives

Dividing by \(\sin^2x\), where \(\sin x\ne0\), gives

The restrictions matter because division by a zero coordinate is not allowed. The addition formulas in Section 15.4 generate useful double-angle formulas:

Using \(\sin^2x+\cos^2x=1\), cosine also has the alternatives \(\cos2x=1-2\sin^2x=2\cos^2x-1\). Choose the version that contains the quantity already present in a problem.

15.8 Differentiation, rates, and small-angle reasoning

The derivative formula \((\sin x)'=\cos x\) depends on radian measure. If an angle is measured in degrees, an extra constant conversion factor appears. This is why calculus and engineering signal models conventionally use radians.

Why This Matters

Rational expressions appear in transfer-like functions and partial fractions. Inverse trigonometric functions recover angles from ratios. Hyperbolic functions arise naturally from exponentials and describe a geometry parallel to, but different from, circle trigonometry.

16.1 Function characteristics and the sigmoid

When a mathematical model is designed from requirements, begin with its desired characteristics: domain, range, symmetry, monotonicity, smoothness, limiting values, roots, and rate of change. The sigmoid

is a useful example. It is defined for every real \(x\), lies strictly between \(0\) and \(1\), approaches \(0\) as \(x\to-\infty\), approaches \(1\) as \(x\to\infty\), and is strictly increasing because

It demonstrates a general design lesson: a function need not be chosen because it looks familiar. Its properties can be checked against the requested behaviour.

16.2 Rational functions and partial fractions

A rational function is \(N(x)/D(x)\) where \(N\) and \(D\) are polynomials and \(D(x)\ne0\). Zeros come from roots of \(N\) not cancelled by \(D\); roots of \(D\) give excluded inputs and may give poles. If the numerator degree is at least the denominator degree, perform polynomial division first.

For distinct linear factors,

Multiplying through by the denominator and choosing convenient \(x\) values finds the coefficients. Repeated factors and irreducible quadratics require expanded templates; do not use the distinct-linear template blindly.

16.3 Inverse trigonometric functions

Trigonometric functions are periodic, so they must be restricted before they can have inverse functions. Standard principal branches are

The notation \(\arcsin x\) avoids confusing an inverse with \(1/\sin x\).

Implicit differentiation of \(\sin y=x\) gives

On the arcsine range, \(\cos y\ge0\), and \(\cos y=\sqrt{1-\sin^2y}=\sqrt{1-x^2}\). Thus

Similarly,

For \(a>0\),

The reciprocal inverse functions require especially careful absolute values. On their standard principal branches,

The \(|x|\) is not optional: \(\sqrt{x^2}=|x|\), not generally \(x\). Different texts make different endpoint choices for these inverse functions; the displayed derivatives are asserted only on intervals where a consistent branch has been chosen.

16.4 Hyperbolic functions

Define

\(\cosh x\) is even and \(\sinh x\) is odd. Their identity is

so \((\cosh t,\sinh t)\) traces the hyperbola \(X^2-Y^2=1\), unlike the circle identity for sine and cosine. Their derivatives are

The inverse functions have logarithmic forms on their principal real domains:

The quotient

is defined for every real \(x\), is odd, and satisfies \(-1<\tanh x<1\). The identity \(\cosh^2x-\sinh^2x=1\) gives

Its inverse has the real domain \(-1<x<1\):

The remaining useful inverse-hyperbolic derivatives are

Consequently, for \(a>0\),

on intervals where the integrands are defined. The latter statement is local: its denominator excludes \(x=\pm a\).

16.5 Inverse functions: branch first, formula second

An inverse function must return one output for each permitted input. Because \(\sin\theta=1/2\) at infinitely many angles, \(\arcsin(1/2)\) cannot mean “all solutions.” It means the one angle in the chosen interval \([-\pi/2,\pi/2]\), namely \(\pi/6\). Solving a trigonometric equation is a different task: after finding a reference angle, all periodic solutions must be listed if the problem requests them.

16.6 Inverse-trigonometric integrals by pattern recognition

The expressions \(a^2-x^2\) and \(a^2+x^2\) signal different geometries. Before using a formula, factor constants until the denominator exactly matches one of the standard patterns.

16.7 Hyperbolic functions are exponential combinations

Circular functions use a plus sign in \(\sin^2x+\cos^2x=1\) and are periodic. Hyperbolic functions use a minus sign in \(\cosh^2x-\sinh^2x=1\) and are not periodic. From their exponential definitions,

Adding and subtracting those equations recovers the definitions. This makes hyperbolic functions natural whenever a calculation already contains both growing and decaying exponentials.

Why This Matters

An algebraic equation relates quantities at one instant. A differential equation relates a quantity to how it changes. That is the natural language of a capacitor, whose current depends on the time rate of change of voltage, and of any system with memory.

17.1 What an ODE is

An ordinary differential equation (ODE) contains an unknown function of one variable and one or more derivatives. Its order is the highest derivative present. For example,

is first order. A solution is a function \(y(t)\) that makes the equation true on the interval of interest.

An initial condition such as \(y(0)=y_0\) selects one physical response from a family of mathematical solutions. Without enough initial conditions, an ODE usually has arbitrary constants and therefore does not yet describe one unique trajectory.

17.2 Linear first-order equations and integrating factors

The standard linear form is

Define the integrating factor

Because \(\mu'=P\mu\),

Integrating gives

and then divide by the nonzero exponential \(\mu\). For constant \(P=k\), the integrating factor is simply \(e^{kt}\).

17.3 RC circuit response

For a series resistor-capacitor circuit driven by a constant source \(V_s\), KVL and the capacitor relation \(i=C\,dv_C/dt\) lead to

With an initially uncharged capacitor, \(v_C(0)=0\), the solution is

The current is

Every symbol has a role: \(V_s\) is the applied step voltage, \(R>0\) and \(C>0\) set the time constant \(\tau=RC\), and the initial condition determines the transient coefficient. A different initial capacitor voltage produces a different solution.

17.4 Sinusoidal forcing and complex method

An AC-forced first-order equation often contains integrals such as \(\int e^{pt}\cos(\omega t)\,dt\). Euler’s formula lets us compute the real part of \(\int e^{(p+i\omega)t}\,dt\), then take the real component. This method is efficient because differentiation and integration of exponentials preserve their form. The completed response separates into a decaying transient and a persistent sinusoidal steady state.

17.5 Separation, linearity, and what an initial condition selects

Some first-order ODEs are separable, meaning they can be rearranged into a function of \(y\) times \(dy\) equal to a function of \(t\) times \(dt\). For example,

has, where \(y\ne0\),

Integrating gives \(\ln|y|=kt+C\), hence \(y=Ae^{kt}\). The missing zero solution is also contained by choosing \(A=0\). Separation is efficient when available, but a linear equation such as \(y'+P(t)y=Q(t)\) need not be separable. The integrating factor method works for every equation in that standard linear form.

17.6 Integrating factors, derived as a product rule

The integrating factor is sometimes taught as a recipe. Its purpose is to make the left side of a linear ODE into one derivative. Start with

We seek a nonzero function \(\mu(t)\) such that

equals \(\mu y'+\mu P y\). This requires \(\mu'=P\mu\). Dividing by \(\mu\) and integrating gives

Any nonzero constant multiple would work and cancels from the final solution.

17.7 RC response from the circuit law

For a series resistor and capacitor with source \(V_s\), Kirchhoff's voltage law is

The resistor voltage is \(v_R=Ri\), and the capacitor current is \(i=C\,dv_C/dt\) under the passive sign convention. Substitution gives

Dividing by \(RC\) puts the equation in standard form:

The integrating factor is \(e^{t/(RC)}\). Carrying out the method gives the general step response

where \(V_0=v_C(0)\). It passes two physical checks:

17.8 First-order problem-solving strategy

1. Name the independent variable and the unknown function; state an interval.
2. Classify the equation: separable, linear, or neither of these standard forms.
3. For linear equations, identify \(P(t)\) and \(Q(t)\) before forming \(\mu\).
4. Keep the arbitrary constant until every integration is complete.
5. Apply initial data, then substitute the result back into the ODE and inspect \(t=0\) and long-time behaviour.

For a stable constant-coefficient equation \(y'+ky=q\) with \(k>0\), the homogeneous part \(Ce^{-kt}\) decays. The particular constant \(q/k\) is the steady state. This separation is a mathematical result, not an assumption that every real circuit is ideal for all times.

Why This Matters

Two energy-storage elements create richer dynamics. Inductors and capacitors can exchange energy, producing oscillation; resistance removes energy, producing damping. Second-order ODEs classify these behaviours through the roots of one quadratic equation.

18.1 Homogeneous constant-coefficient equations

Consider

Try \(y=e^{st}\). Since \(y'=se^{st}\) and \(y''=s^2e^{st}\),

The exponential never equals zero, so \(s\) must satisfy the characteristic equation

The nature of its roots determines the solution.

• Distinct real roots \(s_1,s_2\): \(y=C_1e^{s_1t}+C_2e^{s_2t}.\)
• Repeated real root \(s\): \(y=(C_1+C_2t)e^{st}.\)
• Complex roots \(\alpha\pm i\beta\): \(y=e^{\alpha t}(C_1\cos\beta t+C_2\sin\beta t).\)

Two initial conditions, typically \(y(0)\) and \(y'(0)\), determine \(C_1\) and \(C_2\).

18.2 RLC damping

For a series RLC natural response, a standard current equation can be written

where

The characteristic roots are

If \(\alpha>\omega_0\), roots are distinct real values: the response is over-damped, non-oscillatory. If \(\alpha=\omega_0\), there is a repeated root: critical damping, the fastest non-oscillatory return under this model. If \(\alpha<\omega_0\), roots are complex and the response is under-damped:

\(\omega_d\) is an oscillation frequency only in the under-damped case. It should not be defined with an absolute value to imply oscillation in an over-damped circuit.

18.3 Non-homogeneous equations

For

the general solution is

where \(y_h\) solves the homogeneous equation and \(y_p\) is one particular solution. For polynomial, exponential, sine, or cosine forcing, the method of undetermined coefficients proposes a trial form whose derivatives have the same family.

If the proposed trial duplicates a term already in \(y_h\), multiply the trial by a sufficient power of \(t\) to make it independent. This resonance correction is essential; otherwise substituted coefficients cannot determine a new solution.

18.4 Physical interpretation

The homogeneous part contains natural modes determined by stored energy and losses. In a stable damped circuit, it decays and becomes the transient. The particular part is driven by the source and becomes the steady-state response. With \(R=0\), the ideal LC equation has undamped sinusoidal solutions; calling it an LC oscillator equation is more precise here than confusing it with the spatial wave-equation PDE.

18.5 Why the characteristic equation supplies all three cases

Trying \(y=e^{st}\) is useful because differentiation only multiplies this trial by \(s\) or \(s^2\). Substitution into a constant-coefficient homogeneous equation therefore reduces a function problem to the algebraic quadratic

If the roots are distinct, \(e^{s_1t}\) and \(e^{s_2t}\) are independent solutions, so their linear combination contains two arbitrary constants. If the root is repeated, simply writing the same exponential twice does not give two independent solutions. The second solution is \(te^{st}\), producing

If roots are \(\alpha\pm i\beta\), Euler's formula turns their exponential solutions into real functions:

Adding and subtracting conjugate solutions gives the real basis \(e^{\alpha t}\cos\beta t\) and \(e^{\alpha t}\sin\beta t\). The envelope \(e^{\alpha t}\) must remain; it describes growth or decay of the oscillation.

18.6 RLC classification from the discriminant

For the RLC characteristic equation

the discriminant is \(4(\alpha^2-\omega_0^2)\). Its sign creates the three physical response classes:

18.7 Non-homogeneous equations: choose, test, correct

For \(Ly=q(t)\), first solve \(Ly=0\) to obtain \(y_h\). Then choose a trial \(y_p\) whose derivatives remain in the same family as \(q(t)\). Substitute the entire trial and equate coefficients. If the trial overlaps \(y_h\), multiply it by \(t\) once for each overlap order; otherwise it cannot create a genuinely new solution.

18.8 A second-order solution checklist

1. Put the ODE in standard form and identify whether it is homogeneous.
2. Form and solve the characteristic equation before writing a solution shape.
3. Select the correct distinct, repeated, or complex-root family.
4. If forced, find one particular solution independent of the homogeneous family.
5. Apply both initial conditions and substitute back into the original equation.
6. Interpret signs of real parts: negative means decay, zero means persistent oscillation, positive means growth.