Questions covered in Standard

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. The value of $\lim_{x \to 0} \frac{\sin 4x}{\sin 2x}$ is
2. The derivative of $e^x \sin x$ is
3. Evaluate $\lim_{x \to 2} \frac{x^3 - 8}{x - 2}$.
4. The derivative of $\log x$ with respect to $x$ is
5. If $f(x) = x^{100} + x^{99} + \dots + x + 1$, then $f'(1)$ is
6. Evaluate $\lim_{x \to 0} \frac{\sqrt{1+x} - 1}{x}$.
7. Find the derivative of $f(x) = \frac{x+1}{x-1}$.
8. Evaluate $\lim_{x \to 0} \frac{1 - \cos x}{x}$.
9. Find the derivative of $f(x) = (x + \sec x)(x - \tan x)$.
10. Evaluate $\lim_{x \to 0} f(x)$, where $f(x) = \begin{cases} 2x+3, & x \le 0 \\ 3(x+1), & x > 0 \end{cases}$.
11. Find the derivative of $\tan x$ from first principles.
12. Case Study: A particle moves along a straight line such that its distance $s$ in meters from a fixed point at time $t$ seconds is given by $s(t) = t^3 - 6t^2 + 9t + 5$.
    (i) Find the velocity $v(t)$ at any time $t$.
    (ii) Find the acceleration $a(t)$ at any time $t$.
    (iii) When is the particle at rest?