L3 Differentiation
Question Bank
The graph below shows the function $y = f(x)$.
For the function above:
(a) What is the value of $f(4)$? State clearly if the value does not exist.
(b) What is the value of $\lim\limits_{x \to 4} f(x)$? State clearly if the value does not exist.
(c) Find the value(s) of $x$ where $f(x)$ is not continuous.
The graph below shows the function $y = f(x)$.
For the function above:
(a) Find the value(s) of $x$ where $f(x)$ is not differentiable.
(b) Find the value(s) of $x$ where $f'(x) = 0$.
(c) What is the value of $\lim\limits_{x \to 3} f(x)$? State clearly if the value does not exist.
The graph below shows the function $y = f(x)$.
For the function above:
(a) Find the value(s) of $x$ where $f(x)$ is continuous but not differentiable.
(b) Find the value(s) of $x$ where $f'(x) = 0$ and $f''(x) < 0$.
(c) What is the value of $\lim\limits_{x \to 4} f(x)$? State clearly if the value does not exist.
(d) What is the value of $f(4)$? State clearly if the value does not exist.
The graph below shows the function $y = f(x)$.
For the function above:
(a) Find the value(s) of $x$ where $f(x)$ is not defined.
(b) What is the value of $\lim\limits_{x \to -5} f(x)$? State clearly if the value does not exist.
(c) What is the value of $\lim\limits_{x \to 3} f(x)$? State clearly if the value does not exist.
(d) Find the value(s) of $x$ where $f'(x) > 0$.
The graph below shows the function $y = f(x)$.
For the function above:
(a) What are the values of $\lim\limits_{x \to 2^-} f(x)$ and $\lim\limits_{x \to 2^+} f(x)$?
(b) What is the value of $\lim\limits_{x \to 2} f(x)$? State clearly if the value does not exist.
(c) What is the value of $f(2)$? State clearly if the value does not exist.
(d) Find the value(s) of $x$ where $f(x)$ is concave up.
The graph below shows the function $y = f(x)$.
For the function above:
(a) Find the value(s) of $x$ where $f(x)$ is not continuous.
(b) Find the value(s) of $x$ where $f(x)$ is continuous but not differentiable.
(c) Find the value(s) of $x$ where $f'(x) = 0$.
(d) What is the value of $f(3)$? State clearly if the value does not exist.
The graph below shows the function $y = f(x)$ for $-6 \le x \le 5$.
For the function above:
(a) What is the value of $f(-2)$? State clearly if the value does not exist.
(b) What is the value of $\lim\limits_{x \to 2} f(x)$? State clearly if the value does not exist.
(c) Find the value(s) of $x$ where $f'(x) < 0$.
(d) Find the value(s) of $x$ where $f''(x) < 0$.
The graph below shows the function $y = f(x)$.
For the function above:
(a) What is the value of $f(6)$? State clearly if the value does not exist.
(b) For what value(s) of $x$ does $f(x)$ not have a limit?
(c) Find the value(s) of $x$ where $f'(x) > 0$.
(d) Find the value(s) of $x$ where $f'(x) = 0$ and $f''(x) < 0$.
(e) Find the value(s) of $x$ where $f(x)$ is continuous but not differentiable.
The graph below shows the function $y = f(x)$.
For the function above:
(a) Find the value(s) of $x$ where $f(x)$ is not continuous.
(b) Find the value(s) of $x$ where $f(x)$ is continuous but not differentiable.
(c) What is the value of $\lim\limits_{x \to -4} f(x)$? State clearly if the value does not exist.
(d) What is the value of $\lim\limits_{x \to 5} f(x)$? State clearly if the value does not exist.
(e) Find the value(s) of $x$ where $f'(x) < 0$.
The graph below shows the function $y = f(x)$.
For the function above:
(a) What is the value of $f(-4)$? State clearly if the value does not exist.
(b) What is the value of $\lim\limits_{x \to -4} f(x)$? State clearly if the value does not exist.
(c) Find the value(s) of $x$ where $f'(x) = 0$.
(d) Find the value(s) of $x$ where $f(x)$ is continuous but not differentiable.
(e) Find the value(s) of $x$ where $f'(x) > 0$.
The function $f$ is defined by
$ \normalsize\small f(x) = \begin{cases} 2\sin x + 1, & x \le 0 \\ (x + 1)^2, & 0 < x < 1 \\ 6 - \dfrac{2}{x}, & x \ge 1 \end{cases} $
(a) Show that $f$ is continuous at $x = 0$ and at $x = 1$.
(b) Determine whether $f$ is differentiable at $x = 0$ and at $x = 1$. Justify your answers using one-sided limits.
You must use calculus and show any derivatives that you need to find when solving this problem.
The function $f$ is defined by
$ \normalsize\small f(x) = \begin{cases} ax^2 + bx, & x \le 2 \\ 6\ln(x - 1) + 4, & x > 2 \end{cases} $
where $a$ and $b$ are constants.
Find the values of $a$ and $b$ for which $f$ is both continuous and differentiable at $x = 2$.
You must use calculus and show any derivatives that you need to find when solving this problem.