Appendices A through R, Glossary, Answer Key reference, Index, and MGU Library Connections — the complete reference section following Chapter 14
This appendix reviews the prerequisite knowledge most important for success in calculus. Students who find any section unfamiliar should review the corresponding chapters in Trigonometry and Precalculus in the MGU Mathematics Series before proceeding.
A function assigns exactly one output to each input. If f is a function and x is an input value, then f(x) is the corresponding output.
Domain — the set of all valid input values.
Range — the set of all possible output values.
Vertical line test — a graph represents a function if and only if every vertical line crosses it at most once.
Composition — (f ∘ g)(x) = f(g(x)). The output of g becomes the input of f.
Inverse functions — f⁻¹ undoes f. If f(a) = b, then f⁻¹(b) = a.
| Function | Shape | Domain | Range |
|---|---|---|---|
| f(x) = c (constant) | Horizontal line | All reals | {c} |
| f(x) = x (linear) | Line through origin | All reals | All reals |
| f(x) = x² (quadratic) | Upward parabola | All reals | [0, ∞) |
| f(x) = x³ (cubic) | S-curve | All reals | All reals |
| f(x) = √x (square root) | Half-curve | [0, ∞) | [0, ∞) |
| f(x) = |x| (absolute value) | V-shape | All reals | [0, ∞) |
| f(x) = 1/x (reciprocal) | Two hyperbola branches | x ≠ 0 | y ≠ 0 |
| f(x) = eˣ (natural exponential) | Increasing curve | All reals | (0, ∞) |
| f(x) = ln x (natural log) | Increasing curve | (0, ∞) | All reals |
| f(x) = sin x | Wave | All reals | [−1, 1] |
| f(x) = cos x | Wave | All reals | [−1, 1] |
| Transformation | Effect |
|---|---|
f(x) + k | Vertical shift up k units |
f(x) − k | Vertical shift down k units |
f(x − h) | Horizontal shift right h units |
f(x + h) | Horizontal shift left h units |
−f(x) | Reflection across the x-axis |
f(−x) | Reflection across the y-axis |
af(x), |a| > 1 | Vertical stretch |
af(x), 0 < |a| < 1 | Vertical compression |
m = (y₂ − y₁)/(x₂ − x₁) = Δy/Δx
Slope-intercept: y = mx + b
Point-slope: y − y₁ = m(x − x₁)
Parallel lines: equal slopes. Perpendicular lines: slopes are negative reciprocals.
Polynomials are defined and continuous everywhere. Rational functions p(x)/q(x) are undefined where q(x) = 0, and can have vertical asymptotes, horizontal asymptotes, holes, and oblique asymptotes.
f(x) = aˣ: growth if a > 1; decay if 0 < a < 1. Natural base: e ≈ 2.71828.
ln(ab) = ln a + ln b
ln(a/b) = ln a − ln b
ln(aʳ) = r ln a
Change of base: log_a(x) = ln x / ln a
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| π/6 | 1/2 | √3/2 | 1/√3 |
| π/4 | √2/2 | √2/2 | 1 |
| π/3 | √3/2 | 1/2 | √3 |
| π/2 | 1 | 0 | undefined |
| π | 0 | −1 | 0 |
| 3π/2 | −1 | 0 | undefined |
| 2π | 0 | 1 | 0 |
Factoring polynomials (common factor, difference of squares, trinomial). Simplifying rational expressions by canceling common factors. Rationalizing numerators and denominators using conjugates. Laws of exponents: xᵐ · xⁿ = xᵐ⁺ⁿ, xᵐ/xⁿ = xᵐ⁻ⁿ, (xᵐ)ⁿ = xᵐⁿ. Fractional exponents: x^(1/n) = ⁿ√x, x^(m/n) = (ⁿ√x)ᵐ. These appear constantly in evaluating limits, differentiating, and integrating.
For each function, the key calculus-relevant features — derivative, integral, and graphical behavior — are noted alongside the shape description.
sin²x + cos²x = 1
1 + tan²x = sec²x
1 + cot²x = csc²x
csc x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x
tan x = sin x / cos x, cot x = cos x / sin x
sin(−x) = −sin x (odd)
cos(−x) = cos x (even)
tan(−x) = −tan x (odd)
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
sin(2x) = 2 sin x cos x
cos(2x) = cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x
sin²x = (1 − cos 2x) / 2 (power-reducing — essential for integration)
cos²x = (1 + cos 2x) / 2 (power-reducing — essential for integration)
| Function | Domain | Range |
|---|---|---|
| arcsin x | [−1, 1] | [−π/2, π/2] |
| arccos x | [−1, 1] | [0, π] |
| arctan x | all reals | (−π/2, π/2) |
lim(x→0) [sin x / x] = 1
lim(x→0) [(1 − cos x) / x] = 0
These limits are the foundation of all trigonometric derivatives. They are proved using the Squeeze Theorem.
All laws below assume that lim(x→a) f(x) = L and lim(x→a) g(x) = M both exist.
| Law | Formula |
|---|---|
| Sum | lim[f + g] = L + M |
| Difference | lim[f − g] = L − M |
| Constant Multiple | lim[c · f] = c · L |
| Product | lim[f · g] = L · M |
| Quotient | lim[f / g] = L / M, provided M ≠ 0 |
| Power | lim[f(x)]ⁿ = Lⁿ |
| Root | lim ⁿ√f(x) = ⁿ√L (L > 0 when n is even) |
lim(x→a) c = c (limit of a constant)
lim(x→a) x = a
lim(x→a) xⁿ = aⁿ
The two-sided limit lim(x→a) f(x) = L exists if and only if both one-sided limits exist and are equal:
lim(x→a⁻) f(x) = L and lim(x→a⁺) f(x) = L
lim(x→∞) 1/xⁿ = 0 for any n > 0
For rational functions, the limit at infinity is determined by the ratio of the leading terms. Compare degrees of numerator and denominator.
If g(x) ≤ f(x) ≤ h(x) near x = a, and lim(x→a) g(x) = lim(x→a) h(x) = L, then lim(x→a) f(x) = L.
Used to prove lim(x→0) [sin x / x] = 1 and other results.
1. f(a) is defined.
2. lim(x→a) f(x) exists.
3. lim(x→a) f(x) = f(a).
If any condition fails, the function has a discontinuity at x = a.
| Type | Description | Graph Appearance |
|---|---|---|
| Removable | Limit exists, but f(a) is undefined or f(a) ≠ limit. | Hole in the graph |
| Jump | One-sided limits exist but are not equal. | Graph jumps vertically |
| Infinite | At least one one-sided limit is infinite. | Vertical asymptote |
| Oscillating | Function oscillates without settling near x = a. | Rapid oscillation (e.g., sin(1/x) near 0) |
Polynomial functions · Rational functions (where denominator ≠ 0) · Radical functions (on their domains) · Trigonometric functions (on their domains) · Exponential functions · Logarithmic functions · Compositions and sums of continuous functions
If f is continuous on [a, b] and k is any value between f(a) and f(b), then there exists at least one value c ∈ (a, b) such that f(c) = k.
The IVT guarantees existence — it does not find the value. A continuous function cannot skip from one value to another without passing through every value in between.
f'(x) = lim(h→0) [f(x + h) − f(x)] / h
| Rule | Formula |
|---|---|
| Constant Rule | d/dx[c] = 0 |
| Power Rule | d/dx[xⁿ] = nxⁿ⁻¹ |
| Constant Multiple Rule | d/dx[c·f(x)] = c·f'(x) |
| Sum Rule | d/dx[f + g] = f' + g' |
| Difference Rule | d/dx[f − g] = f' − g' |
| Product Rule | d/dx[fg] = f'g + fg' |
| Quotient Rule | d/dx[f/g] = (f'g − fg') / g² |
| Chain Rule | d/dx[f(g(x))] = f'(g(x)) · g'(x) |
| Function | Derivative |
|---|---|
sin x | cos x |
cos x | −sin x |
tan x | sec²x |
sec x | sec x tan x |
csc x | −csc x cot x |
cot x | −csc²x |
eˣ | eˣ |
aˣ | aˣ ln a |
ln x | 1/x |
log_a(x) | 1/(x ln a) |
arcsin x | 1/√(1 − x²) |
arccos x | −1/√(1 − x²) |
arctan x | 1/(1 + x²) |
f'(x): first derivative
f''(x): second derivative
f⁽ⁿ⁾(x): nth derivative
Leibniz notation: d²y/dx², d³y/dx³
d/dx[c] = 0
d/dx[x] = 1
d/dx[x²] = 2x
d/dx[x³] = 3x²
d/dx[xⁿ] = nxⁿ⁻¹
d/dx[x⁻¹] = −x⁻² = −1/x²
d/dx[√x] = 1/(2√x)
d/dx[x^(1/n)] = (1/n)·x^((1/n)−1)
d/dx[eˣ] = eˣ
d/dx[e^(kx)] = ke^(kx)
d/dx[aˣ] = aˣ ln a
d/dx[ln x] = 1/x
d/dx[ln|x|] = 1/x
d/dx[ln(g(x))] = g'(x)/g(x)
d/dx[log_a(x)] = 1/(x ln a)
d/dx[sin x] = cos x
d/dx[cos x] = −sin x
d/dx[tan x] = sec²x
d/dx[sec x] = sec x tan x
d/dx[csc x] = −csc x cot x
d/dx[cot x] = −csc²x
d/dx[arcsin x] = 1/√(1 − x²)
d/dx[arccos x] = −1/√(1 − x²)
d/dx[arctan x] = 1/(1 + x²)
Related rates problems describe two or more quantities that are both changing with respect to time. The goal is to find the rate of change of one quantity given the rate of change of another.
Right triangle (a² + b² = c²): 2a·(da/dt) + 2b·(db/dt) = 2c·(dc/dt)
Circle area (A = πr²): dA/dt = 2πr·(dr/dt)
Sphere volume (V = 4/3·πr³): dV/dt = 4πr²·(dr/dt)
Cylinder (V = πr²h): dV/dt = πr²·(dh/dt) + 2πrh·(dr/dt)
Cone (V = 1/3·πr²h): Differentiate using the product rule after relating r and h.
Substituting a specific value of r, h, or x before differentiating — differentiate first, then substitute. Forgetting to differentiate every changing term. Ignoring the sign of a rate (a decreasing quantity has a negative rate). Misidentifying which rate is given and which is unknown. Forgetting units on rates — always express as a quantity per unit of time.
If Q is continuous on [a, b]:
1. Find all critical numbers of Q on (a, b).
2. Evaluate Q at all critical numbers and at both endpoints a and b.
3. The largest value is the absolute maximum. The smallest is the absolute minimum.
| Situation | Objective Function |
|---|---|
| Rectangle with fixed perimeter P | Maximize A = lw, constrained by 2l + 2w = P |
| Box with no lid, square base | Minimize surface area or maximize volume |
| Cylindrical can, fixed volume V | Minimize surface area 2πr² + 2πrh |
| Distance minimization | Minimize d² = (x − a)² + (y − b)² |
| Revenue and cost | Maximize profit P(x) = R(x) − C(x) |
An antiderivative of f(x) is any function F(x) such that F'(x) = f(x). The family of all antiderivatives is the indefinite integral: ∫f(x)dx = F(x) + C.
| Rule | Formula |
|---|---|
| Constant | ∫k dx = kx + C |
| Power Rule (n ≠ −1) | ∫xⁿ dx = xⁿ⁺¹/(n+1) + C |
| Constant Multiple | ∫k·f(x) dx = k·∫f(x) dx |
| Sum | ∫[f + g] dx = ∫f dx + ∫g dx |
| Difference | ∫[f − g] dx = ∫f dx − ∫g dx |
The power rule ∫xⁿ dx = xⁿ⁺¹/(n+1) + C does not apply when n = −1. That case is handled by the natural log rule: ∫(1/x) dx = ln|x| + C.
∫eˣ dx = eˣ + C
∫e^(kx) dx = (1/k)e^(kx) + C
∫aˣ dx = aˣ/ln a + C (a > 0, a ≠ 1)
∫(1/x) dx = ln|x| + C
∫sin x dx = −cos x + C
∫cos x dx = sin x + C
∫sec²x dx = tan x + C
∫csc²x dx = −cot x + C
∫sec x tan x dx = sec x + C
∫csc x cot x dx = −csc x + C
∫1/√(1 − x²) dx = arcsin x + C
∫1/(1 + x²) dx = arctan x + C
∫x dx = x²/2 + C
∫x² dx = x³/3 + C
∫x³ dx = x⁴/4 + C
∫√x dx = (2/3)x^(3/2) + C
∫1/√x dx = 2√x + C
∫1/x² dx = −1/x + C
∫eˣ dx = eˣ + C
∫e^(2x) dx = (1/2)e^(2x) + C
∫2ˣ dx = 2ˣ/ln 2 + C
∫(1/x) dx = ln|x| + C
∫1/(ax + b) dx = (1/a)ln|ax + b| + C
∫sin x dx = −cos x + C
∫cos x dx = sin x + C
∫tan x dx = ln|sec x| + C
∫sec²x dx = tan x + C
∫csc²x dx = −cot x + C
∫₀¹ xⁿ dx = 1/(n+1)
∫₀^π sin x dx = 2
∫₀^π cos x dx = 0
∫₀¹ eˣ dx = e − 1
∫₁ᵉ (1/x) dx = 1
Width of each subinterval: Δx = (b − a) / n
Partition points: x₀ = a, x₁ = a + Δx, x₂ = a + 2Δx, …, xₙ = b
| Type | Formula | Sample Point |
|---|---|---|
| Left-endpoint | Lₙ = Σ f(xₖ₋₁)·Δx, k=1 to n | Left endpoint of each subinterval |
| Right-endpoint | Rₙ = Σ f(xₖ)·Δx, k=1 to n | Right endpoint of each subinterval |
| Midpoint | Mₙ = Σ f((xₖ₋₁ + xₖ)/2)·Δx | Midpoint — generally most accurate |
lim(n→∞) Rₙ = lim(n→∞) Lₙ = lim(n→∞) Mₙ = ∫ₐᵇ f(x) dx
All three Riemann sums converge to the definite integral provided f is integrable on [a, b].
Σᵢ₌₁ⁿ c = cn
Σᵢ₌₁ⁿ k = n(n+1)/2
Σᵢ₌₁ⁿ k² = n(n+1)(2n+1)/6
Σᵢ₌₁ⁿ k³ = [n(n+1)/2]²
When f(x) > 0: the definite integral gives positive area. When f(x) < 0: the definite integral gives negative area (below the x-axis). Net area = area above axis − area below axis. Total area = area above axis + area below axis. To find total area, split the integral at each zero and take the absolute value of each piece.
Let f be continuous on [a, b]. The accumulation function is: F(x) = ∫ₐˣ f(t) dt
F(x) represents the net signed area under f from a to x. As x increases, F tracks how area accumulates.
If f is continuous on [a, b] and F(x) = ∫ₐˣ f(t) dt, then F is differentiable and:
F'(x) = f(x)
The derivative of an accumulation function returns the original integrand. Rate and accumulation are inverse processes.
With chain rule (extended form): d/dx [∫ₐ^(g(x)) f(t) dt] = f(g(x)) · g'(x)
If f is continuous on [a, b] and F is any antiderivative of f, then:
∫ₐᵇ f(x) dx = F(b) − F(a) = [F(x)]ₐᵇ
To evaluate a definite integral: find any antiderivative of the integrand and evaluate the difference at the endpoints.
∫ₐᵇ F'(x) dx = F(b) − F(a)
The integral of a rate of change gives total net change in the quantity. This is FTC Part 2 stated in terms of what the function and its derivative represent.
f_avg = (1/(b − a)) · ∫ₐᵇ f(x) dx
If f is continuous on [a, b], there exists c ∈ (a, b) such that f(c) = f_avg. A continuous function must actually equal its average value somewhere on the interval.
If f(x) ≥ g(x) on [a, b]: A = ∫ₐᵇ [f(x) − g(x)] dx
Always integrate top minus bottom (or right minus left when integrating with respect to y). Find intersection points by setting f(x) = g(x) and solving for x. If the curves cross more than once, split the integral at each crossing.
V = ∫ₐᵇ A(x) dx where A(x) is the cross-sectional area at position x
V = π ∫ₐᵇ [f(x)]² dx (region between y = f(x) and x-axis, rotated about x-axis)
V = π ∫ₐᵇ ([f(x)]² − [g(x)]²) dx (outer radius f, inner radius g, rotated about x-axis)
V = 2π ∫ₐᵇ x · f(x) dx (region bounded by y = f(x), rotated about y-axis)
| Method | Revolution Axis | Formula Structure |
|---|---|---|
| Disk | x-axis | π ∫[f]² dx |
| Washer | x-axis | π ∫([f]² − [g]²) dx |
| Shell | y-axis | 2π ∫ x·f(x) dx |
A differential equation is an equation that relates an unknown function to its derivatives. The unknown is a function, not a number. A solution is a function y = f(x) that satisfies the equation on some interval. A general solution contains an arbitrary constant C; an initial condition determines the specific value of C.
A differential equation is separable if it can be written as: dy/dx = g(x) · h(y)
Separate variables: dy/h(y) = g(x) dx
Then integrate both sides: ∫1/h(y) dy = ∫g(x) dx
dy/dt = ky → y = Ce^(kt)
k > 0: exponential growth. k < 0: exponential decay. C = y(0) is the initial value.
| Model | Formula |
|---|---|
| Population growth | P(t) = P₀e^(kt) |
| Radioactive decay | A(t) = A₀e^(−kt) |
| Continuous compound interest | A(t) = Pe^(rt) |
| Newton's Law of Cooling | T(t) = T_env + (T₀ − T_env)e^(−kt) |
A slope field draws short line segments at each point (x, y) with slope equal to dy/dx = f(x, y). Together, these segments show the direction of all solution curves. A specific solution curve follows the flow of the slope field. Slope fields are used to visualize solutions when exact formulas are difficult or impossible to find.
A sequence is an ordered list a₁, a₂, a₃, …, aₙ, … A sequence converges to L if lim(n→∞) aₙ = L. If no finite limit exists, the sequence diverges.
An infinite series is the sum of all terms of a sequence: Σaₙ = a₁ + a₂ + a₃ + … A series converges if the sequence of partial sums Sₙ = Σₖ₌₁ⁿ aₖ converges as N → ∞.
Form: Σₙ₌₀^∞ arⁿ = a + ar + ar² + …
Converges if and only if |r| < 1, with sum: a / (1 − r)
If |r| ≥ 1, the series diverges.
A power series centered at x = a is: Σₙ₌₀^∞ cₙ(x − a)ⁿ = c₀ + c₁(x−a) + c₂(x−a)² + … It converges for x values within its radius of convergence. Power series allow functions to be expressed as infinite polynomials.
Pₙ(x) = f(a) + f'(a)(x−a) + f''(a)/2!(x−a)² + … + f⁽ⁿ⁾(a)/n!(x−a)ⁿ
When a = 0: Maclaurin polynomial.
eˣ = 1 + x + x²/2! + x³/3! + …
sin x = x − x³/3! + x⁵/5! − …
cos x = 1 − x²/2! + x⁴/4! − …
1/(1−x) = 1 + x + x² + x³ + … (|x| < 1)
Convergence tests, radius of convergence, Taylor series derivation, and error bounds are covered fully in Calculus II in the MGU Mathematics Series.
Technology should support understanding, not replace reasoning. When technology produces a number, graph, or approximation, students should still explain what it means in context.
| Platform | Primary Use |
|---|---|
| Desmos | Graphing, exploring transformations, area shading, regression |
| GeoGebra | Calculus visualizations, slope fields, 3D graphing |
| WolframAlpha | Evaluating derivatives, integrals, limits, series; symbolic CAS |
| TI-84 / TI-Nspire | Standard graphing calculator for class and standardized testing |
| Python (NumPy/Matplotlib) | Numerical computation and visualization for computing-oriented students |
Use this checklist to confirm readiness for Calculus II in the MGU Mathematics Series. Each item corresponds to skills developed in this textbook.
Integration techniques: integration by parts, trigonometric integrals, trigonometric substitution, partial fractions, improper integrals. Further applications: arc length, surface area, work, hydrostatic pressure. Full sequence and series theory: convergence tests, power series, Taylor and Maclaurin series, error bounds. Parametric equations and polar coordinates. Introduction to multivariable functions as a bridge to Calculus III.
This glossary defines the major terms introduced in Calculus. Entries are listed alphabetically. For fuller explanations, return to the chapter where each term first appears.
The full worked answer key for all practice problems, concept checks, error analysis exercises, and chapter checkpoints is published separately as the Calculus Answer Key and Solutions Manual in the MGU Mathematics Series.
Students working independently may check odd-numbered answers using the brief answer table distributed with the student edition. Full solutions support independent learning when students review worked examples, attempt problems, and then study the solution process step by step.
Section numbers are listed. Appendix references use the letter prefix (e.g., App. F). Glossary references are noted as Glos.
This textbook is part of the MGU Library Edition of the Mathematics Series. It is designed for independent learners, classroom instruction, and library reference. It follows Trigonometry and Precalculus and precedes Calculus II in the MGU mathematics pathway. Students who have completed all fourteen chapters and reviewed the appendices in this back matter are ready for the full course in Calculus II.