Q: If $f(x)$ is defined as $f(x) = \begin{cases} x^2, & x < 0 \\ 2x, & x \geq 0 \end{cases}$, is $f(x)$ differentiable at $x = 0$?

No, because the left and right derivatives at $x = 0$ are not equal

Question 1

Suppose $f(x)$ is continuous at $x = 2$ but not differentiable at $x = 2$. Which of the following could be true?

Accepted Answer

$f(x)$ has a sharp corner at $x = 2$

Question 2

Given $f(x) = x^3 + 5x^2 + 10x + 5$, what is $f'(x)$?

Accepted Answer

$3x^2 + 10x + 10$

Question 3

The average rate of change of a function $f(x)$ from $x = 0$ to $x = 1$ is given by:

Accepted Answer

$\dfrac{f(1) - f(0)}{1 - 0}$

Question 4

If $f(x) = e^{-\cos(x)}$, what is $f'(x)$?

Accepted Answer

$e^{-\cos(x)} \sin(x)$

Question 5

A medication's concentration in the bloodstream is modeled by $C(t) = \sqrt{t} - t^2$. What is the instantaneous rate of change of concentration at $t = 4$ hours?

Accepted Answer

$\dfrac{1}{2\sqrt{4}} - 2 \cdot 4 = \dfrac{1}{4} - 8 = -\dfrac{31}{4}$

Question 6

Which of the following statements about the relationship between a function and its derivative is correct?

Accepted Answer

The graph of $f'(x)$ gives the slope of the tangent line to $f(x)$ at each point.

Question 7

Find the derivative of $g(x) = \dfrac{x}{x-3}$.

Accepted Answer

$\dfrac{-3}{(x-3)^2}$

Question 8

A population of bacteria grows according to $P(t) = 3e^{2t}$. What is the rate of growth at $t = 1$ hour?

Accepted Answer

$6e^{2}$

Question 9

If $f(x)$ is not continuous at $x = a$, which of the following must be true?

Accepted Answer

$f(x)$ is not differentiable at $x = a$

Question 10

Given $h(t) = t^{3/2} - 5t^{1/2}$, what is $h'(t)$?

Accepted Answer

$\dfrac{3}{2} t^{1/2} - \dfrac{5}{2} t^{-1/2}$

Question 11

A car's position is given by $s(t) = t^3 - t$. What is the velocity of the car at $t = 2$ seconds?

Accepted Answer

$11$

Question 12

If $f(x)$ is continuous at $x = 0$ and $\lim_{x 	o 0} f(x) = 2$, which of the following must be true?

Accepted Answer

$f(0) = 2$

Question 13

Find the derivative of $f(t) = (5 - t^4)^5 (5 - 4t^3)$.

Question 14

If $f(x) = e^{1-x}$, what is $f'(x)$?

Accepted Answer

$-e^{1-x}$

Question 15

Suppose $y = 4t + 3$ is the equation of a tangent line to $f(t)$ at $t = 1$. What is $f(1)$?

Accepted Answer

$7$

Question 16

If $f(x)$ is differentiable at $x = a$, which of the following must be true?

Accepted Answer

$f(x)$ is continuous at $x = a$

Question 17

Given $f(x) = [\sin(x^2)]^5$, what is $f'(x)$?

Accepted Answer

$5 [\sin(x^2)]^4 \cos(x^2) \cdot 2x$

Question 18

A function $f(x)$ has $\lim_{x 	o 2^-} f(x) = 4$, $\lim_{x 	o 2^+} f(x) = 0$, and $f(2) = 4$. Which of the following is true?

Accepted Answer

$f(x)$ is not continuous at $x = 2$

Question 19

The instantaneous rate of change of a function at a point is given by:

Accepted Answer

The slope of the tangent line at that point

Question 20

If $f(x)$ is defined as $f(x) = \begin{cases} x^2, & x < 0 \ 2x, & x \geq 0 \end{cases}$, is $f(x)$ differentiable at $x = 0$?

Accepted Answer

No, because the left and right derivatives at $x = 0$ are not equal

Calculus 1A: Limits, Derivatives, and Rates of Change – Study Notes

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