Q: Which of the following limits defines the derivative of $f(x)$ at $x = a$?

$\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Question 1

Which of the following best describes the concept of the derivative at a point $x = a$ for a function $f(x)$?

Accepted Answer

The slope of the tangent line to the curve $y = f(x)$ at $x = a$.

Question 2

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

Accepted Answer

$f'(x) = 3x^2 - 4x + 1$

Question 3

Which of the following limits defines the derivative of $f(x)$ at $x = a$?

Accepted Answer

$\displaystyle \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$

Question 4

If $y = \sin(x^2)$, what is $\frac{dy}{dx}$ using the chain rule?

Accepted Answer

$2x \cos(x^2)$

Question 5

A ball is thrown upward with velocity $v(t) = 20 - 9.8t$. At what time does the ball reach its maximum height?

Accepted Answer

$t = 2.04$ seconds

Question 6

Which of the following is the correct application of the product rule for differentiation for $f(x) = u(x)v(x)$?

Accepted Answer

$f'(x) = u'(x)v(x) + u(x)v'(x)$

Question 7

Find the equation of the tangent line to $y = x^2 + 2x$ at $x = 1$.

Accepted Answer

$y = 4x - 1$

Question 8

If $f(x) = \ln(x)$, what is $f'(x)$?

Accepted Answer

$\frac{1}{x}$

Question 9

Which of the following statements about continuity is true?

Accepted Answer

A function is continuous at $x = a$ if $\displaystyle \lim_{x 	o a} f(x) = f(a)$.

Question 10

Suppose $f(x)$ has a local maximum at $x = c$. Which of the following must be true?

Accepted Answer

$f'(c) = 0$

Question 11

If $f(x) = e^{2x}$, what is $f''(x)$?

Accepted Answer

$4e^{2x}$

Question 12

Which of the following is the correct derivative of $y = \frac{x^2 + 1}{x}$ using the quotient rule?

Accepted Answer

$\frac{x^2 - 1}{x^2}$

Question 13

A company’s profit $P(x)$ in thousands of dollars is given by $P(x) = -2x^2 + 12x - 9$, where $x$ is the number of units sold in hundreds. What is the maximum profit?

Accepted Answer

$15$ thousand dollars

Question 14

Evaluate $\displaystyle \lim_{x 	o 0} \frac{\sin x}{x}$.

Accepted Answer

$1$

Question 15

Which of the following is the correct application of the chain rule for $y = \cos(3x)$?

Accepted Answer

$y' = -3\sin(3x)$

Question 16

If $f(x) = x^4 - 4x^3 + 6x^2$, find all critical points.

Accepted Answer

$x = 0, 1, 2$

Question 17

Which of the following best describes the Mean Value Theorem for derivatives?

Accepted Answer

If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

Question 18

A medication is administered and its concentration in the bloodstream is modeled by $C(t) = 10e^{-0.5t}$, where $t$ is in hours. What is the rate of change of the concentration at $t = 2$ hours?

Accepted Answer

$-3.68$ units/hour

Question 19

Which of the following is the correct derivative of $y = 	an(x)$?

Accepted Answer

$\sec^2(x)$

Question 20

If $f(x) = x^2$ and $g(x) = \sqrt{x}$, what is $\frac{d}{dx}[f(g(x))]$?

Accepted Answer

$x^{1/2}$

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