Questions covered in Advanced

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 $\lim_{x \to 0} \frac{1 - \cos 2x}{x^2}$ is
2. The derivative of $e^x \sin x$ is
3. Evaluate $\lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}$.
4. If $f(x) = \frac{x-a}{x-b}$, then $f'(x)$ is
5. $\lim_{x \to 0} \frac{\log(1+x)}{x}$ is equal to
6. Evaluate $\lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{x}$.
7. Differentiate $y = \frac{ax + b}{cx + d}$ with respect to $x$.
8. Evaluate $\lim_{x \to \pi} \frac{\sin(\pi - x)}{\pi(\pi - x)}$.
9. Find the derivative of $f(x) = \frac{x}{1 + \tan x}$.
10. Evaluate $\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$.
11. Find the derivative of $\tan x$ from first principles.
12. Case Study: The path of a projectile is given by the function $f(x) = -x^2 + 4x + 5$. The derivative $f'(x)$ represents the slope of the tangent to the curve at any point $x$.
    (i) Find the slope of the tangent at $x = 1$.
    (ii) Find the point where the tangent is parallel to the x-axis.
    (iii) Find the equation of the tangent at $x = 3$.