Year11 MATH 3-2-2 Introduction to Differential Calculus warm-up workbook

Unit 1

Focus: Limits, first principles, and basic power rule.

1. Evaluate $\lim_{x \to 3} (x^2 – 2)$.
2. Differentiate $f(x) = x^5$ with respect to $x$.
3. Find the derivative of $y = 4x^3 – 2x + 7$.
4. Differentiate $f(x) = \frac{1}{x^2}$ (Hint: use index laws).
5. Use the Power Rule to find the gradient of $y = x^2$ at $x = 3$.
6. Find $\frac{dy}{dx}$ if $y = \sqrt{x}$.
7. If $f(x) = 2x(x + 3)$, find $f'(x)$.
8. Find the value of $x$ where the derivative of $y = x^2 – 6x$ is zero.
9. Differentiate $y = \frac{3x^4 – 2x}{x}$.
10. State the formula for the derivative from first principles.

Answers:

1. 7 | 2. $5x^4$ | 3. $12x^2 – 2$ | 4. $-2x^{-3}$ or $-\frac{2}{x^3}$ | 5. 6 | 6. $\frac{1}{2\sqrt{x}}$ | 7. $4x + 6$ | 8. $x = 3$ | 9. $9x^2$ | 10. $f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$.