Questions covered in Basic

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 $\lim_{x \to 1} [x^3 - x^2 + 1]$ is
2. $\lim_{x \to 0} \frac{\sin x}{x}$ is equal to
3. The derivative of $f(x) = x$ is
4. The value of $\lim_{x \to 2} \frac{x^2-4}{x-2}$ is
5. The derivative of $\sin x$ with respect to $x$ is
6. Evaluate $\lim_{x \to 1} \frac{x^{15}-1}{x^{10}-1}$.
7. Find the derivative of $f(x) = x^2 - 2$ at $x = 10$.
8. Find the derivative of $f(x) = \frac{1}{x}$.
9. Find the derivative of $f(x) = (x+1)(x-2)$ using the product rule.
10. Evaluate $\lim_{x \to 0} \frac{\sin ax}{bx}$.
11. Find the derivative of $f(x) = x^2$ from first principles.
12. Case Study: The concept of limits is fundamental to calculus. Consider the function $f(x) = \frac{x^2-9}{x-3}$.
    (i) What is the value of the function at $x=3$?
    (ii) Find the left hand limit as $x \to 3$.
    (iii) Find the limit of $f(x)$ as $x \to 3$.