Prerequisites

This book assumes a working command of precalculus. If you are unsure whether your background is sufficient, take the self-assessment below. If you score 80% or above, you are ready. If you score lower, work through Appendix A — Precalculus Review before starting Chapter 1.

What You Need to Know

Algebra (Required)

You should be able to:

  • Factor polynomials (quadratics, special forms like $a^2 - b^2$, grouping)
  • Simplify rational expressions
  • Solve linear and quadratic equations
  • Manipulate exponents and radicals (including fractional exponents like $x^{1/2}$)
  • Work with absolute value and inequalities
  • Use logarithm and exponential properties: $\log(ab) = \log a + \log b$, $e^{\ln x} = x$, etc.

Trigonometry (Required)

You should know:

  • The unit circle: values of $\sin$, $\cos$, $\tan$ at $0$, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$, $\pi$, $3\pi/2$
  • Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$
  • Angle-sum identities: $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$ and similar
  • Graphs of $\sin x$, $\cos x$, $\tan x$ — including periods, amplitudes, asymptotes
  • Inverse trig functions: $\arcsin$, $\arccos$, $\arctan$ and their ranges

Functions (Required)

You should understand:

  • Domain and range
  • Function composition $(f \circ g)(x) = f(g(x))$
  • Inverse functions
  • Even and odd functions
  • Polynomial, rational, exponential, logarithmic, and trigonometric functions
  • Function transformations: shifts, reflections, stretches

Coordinate Geometry (Required)

You should be able to:

  • Write equations of lines (point-slope, slope-intercept, two-point forms)
  • Write equations of circles
  • Understand parabolas $y = ax^2 + bx + c$
  • Plot points and curves in the $xy$-plane

Not Required

You do not need:

  • Any prior calculus
  • Any programming experience (Python is introduced gradually starting Chapter 2)
  • Linear algebra (introduced when needed in Part VI; no prior matrices required)
  • Any specific software (free Python is sufficient)

Self-Assessment

Try these problems. Solutions are at the end. Allow yourself 30 minutes.

  1. Simplify: $\dfrac{x^2 - 4}{x^2 - 5x + 6}$

  2. Solve for $x$: $2x^2 - 5x + 1 = 0$ (use the quadratic formula)

  3. Evaluate: $\ln(e^3) + \log_{10}(1000)$

  4. Find the exact value of $\sin(7\pi/6)$.

  5. If $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, compute $(f \circ g)(2)$.

  6. Find the inverse of $f(x) = 3x - 7$.

  7. Sketch the graph of $y = -(x-2)^2 + 4$. What is the vertex? Where does it cross the $x$-axis?

  8. Solve for $x$: $e^{2x} = 7$ (express in terms of $\ln$).

  9. Simplify: $\sin(x)\cos(x) + \cos(x)\sin(x)$ using a known identity.

  10. Find the equation of the line through $(2, 3)$ and $(5, -1)$.

Solutions

  1. $\dfrac{x+2}{x-3}$ (factor: $(x-2)(x+2)$ over $(x-2)(x-3)$, cancel $x-2$)

  2. $x = \dfrac{5 \pm \sqrt{17}}{4}$

  3. $3 + 3 = 6$

  4. $-1/2$ (in the third quadrant, $\sin$ is negative; reference angle $\pi/6$)

  5. $(f \circ g)(2) = f(g(2)) = f(1) = 2$

  6. $f^{-1}(x) = \dfrac{x+7}{3}$

  7. Downward parabola, vertex at $(2, 4)$, $x$-intercepts at $x = 0$ and $x = 4$

  8. $x = \dfrac{\ln 7}{2}$

  9. $\sin(x)\cos(x) + \cos(x)\sin(x) = 2\sin(x)\cos(x) = \sin(2x)$

  10. $y - 3 = -\dfrac{4}{3}(x - 2)$, or equivalently $y = -\dfrac{4}{3}x + \dfrac{17}{3}$

Scoring

  • 8–10 correct: You are ready. Begin Chapter 1.
  • 5–7 correct: Review Appendix A in the areas where you missed problems, then begin.
  • Below 5 correct: Work through Appendix A thoroughly. Consider also reviewing the relevant Khan Academy precalculus units (free at khanacademy.org). Begin Chapter 1 when you score 8/10 on a re-take.

Mathematical Maturity

Beyond specific content, calculus requires mathematical maturity — comfort with abstraction, careful reading of dense text, and willingness to manipulate symbols without immediately seeing the answer.

You will develop this maturity as you read. It is not a prerequisite. It is an outcome. The book is designed to build maturity gradually; chapter 1 reads at a roughly high-school level, and by chapter 30 you will be reading material that would have been incomprehensible to your pre-calculus self.

That progression is the entire point.