Questions covered in Hard

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 $\lim_{x \to 0} \frac{1 - \cos x \cos 2x}{x^2}$ is
2. The derivative of $f(x) = |x-1|$ at $x=1$ is
3. Evaluate $\lim_{x \to \pi/2} \frac{\tan 2x}{x - \pi/2}$.
4. If $y = \frac{1}{ax+b}$, then $y'$ is
5. $\lim_{x \to 0} \frac{e^x - e^{-x}}{x}$ is equal to
6. Evaluate $\lim_{x \to 0} \frac{\sqrt{1+x+x^2} - 1}{x}$.
7. Find the derivative of $y = \frac{\sin x + \cos x}{\sin x - \cos x}$.
8. Evaluate $\lim_{x \to 0} \frac{x(e^x - 1)}{1 - \cos x}$.
9. Find the derivative of $f(x) = (x + \sec x)(x - \tan x)$.
10. Evaluate $\lim_{x \to 0} \frac{1 - \cos 2x}{3 \tan^2 x}$.
11. Find the derivative of $\sec x$ from first principles.
12. Case Study: A particle moves along a straight line such that its position $s(t)$ at time $t$ is given by $s(t) = t^3 - 9t^2 + 24t$.
    (i) Find the velocity $v(t)$ at any time $t$.
    (ii) Find the interval of time when the velocity is negative.
    (iii) Find the acceleration at $t=2$.