Chapter 0 — Pre-Calculus Refresher
Five minutes of diagnosis, then a focused review of the algebra, functions and trigonometry that calculus runs on.
Calculus asks two new questions — how fast is something changing? and how much has accumulated? — but it answers them with old tools. Every limit you evaluate needs factoring; every derivative needs function notation and exponent laws; every trigonometric derivative needs the unit circle. Students rarely struggle with the calculus idea; they struggle with the algebra underneath it.
This chapter is a short, focused refresher. Take the diagnostic below first: it takes about five minutes and tells you exactly which parts of the review to study.
Numbers & intervals→Functions→Graphs & transformations→Composition→Lines & quadratics→Factoring→Exponents & logs→Inverses→Trigonometry
🎯 By the end of this chapter you can…
- find the domain and range of a function and read them from a graph;
- transform, combine and decompose functions (inner and outer functions);
- factor, simplify fractions and rationalise with conjugates;
- use the laws of exponents and logarithms and solve exponential equations;
- work in radians, use the unit circle, exact values and the key identities.
Diagnostic — what should you review?
Answer all 12 questions. Each one points to a topic in 0.1 Pre-Calculus Review.
Q1. The domain of $f(x)=\sqrt{x-4}$ is
Q2. $y=(x+1)^2-2$ is $y=x^2$ shifted
Q3. If $f(x)=x^2$ and $g(x)=x+1$, then $f\big(g(2)\big)=$
Q4. The slope of the line through $(0,1)$ and $(2,7)$ is
Q5. The solutions of $x^2-x-6=0$ are
Q6. $\dfrac{x^2-1}{x-1}$ simplifies (for $x\ne1$) to
Q7. $\left(x^{2}\right)^{3}\cdot x^{-4}=$
Q8. $\ln\left(e^{5}\right)=$
Q9. The inverse of $f(x)=2x+6$ is
Q10. $\sin\tfrac\pi6=$
Q11. $180^\circ$ in radians is
Q12. $\sin^2x+\cos^2x=$