Q: Which of the following is the correct application of the chain rule for $\frac{d}{dx} \cos^2(1 + \theta)$?

$-2\cos(1 + \theta)\sin(1 + \theta)$

Question 1

Which of the following is the correct derivative of the function $f(x) = e^{x^2}$ with respect to $x$?

Accepted Answer

$2x e^{x^2}$

Question 2

A population of bacteria grows according to the function $P(t) = 500e^{0.2t}$. What is the instantaneous rate of growth of the population at $t = 5$ hours?

Accepted Answer

$P'(5) = 100e$

Question 3

Evaluate the definite integral $\int_0^{\pi/3} \sin(2x) dx$.

Accepted Answer

$\frac{1}{2}$

Question 4

Which of the following is the correct indefinite integral of $f(x) = x^4 + 3x - 10\sqrt{x}$?

Accepted Answer

$\frac{x^5}{5} + \frac{3x^2}{2} - \frac{20}{3}x^{3/2} + C$

Question 5

Given $f(x) = x^2 + 4x + 2$, what are the roots of $f(x)$?

Accepted Answer

$x = -2 \pm \sqrt{2}$

Question 6

Which of the following is the correct application of the chain rule for $\frac{d}{dx} \cos^2(1 + 	heta)$?

Accepted Answer

$-2\cos(1 + 	heta)\sin(1 + 	heta)$

Question 7

Evaluate $\int \sec x 	an x \, dx$.

Accepted Answer

$\sec x + C$

Question 8

A ball is thrown upward and its height at time $t$ seconds is given by $h(t) = -16t^2 + 64t + 80$. What is the maximum height reached by the ball?

Accepted Answer

$144$ feet

Question 9

Which of the following is the correct value of $\cos(5\pi/4)$?

Accepted Answer

$-\frac{\sqrt{2}}{2}$

Question 10

Evaluate $\int_0^{\pi/3} 7\sin(x)\cos(x) dx$ using the double angle formula.

Accepted Answer

$\frac{7}{4}$

Question 11

Which of the following is the correct derivative of $f(x) = \ln(x^2 + 1)$?

Accepted Answer

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

Question 12

Evaluate $\int x e^x dx$.

Accepted Answer

$(x - 1)e^x + C$

Question 13

Which of the following is the correct value of $\sin(2\pi/3)$?

Accepted Answer

$\frac{\sqrt{3}}{2}$

Question 14

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

Accepted Answer

$2\sin x \cos x$

Question 15

Evaluate $\int \frac{1}{x^2 + 4} dx$.

Accepted Answer

$\frac{1}{2} \arctan\left(\frac{x}{2}\right) + C$

Question 16

Which of the following is the correct value of $	an(\pi/4)$?

Accepted Answer

$1$

Question 17

A medication is administered such that its concentration in the bloodstream at time $t$ hours is $C(t) = 10e^{-0.5t}$. What is the rate of change of concentration at $t = 2$ hours?

Accepted Answer

$-5e^{-1}$

Question 18

Evaluate $\int \frac{x-1}{\sqrt{x^2+1}} dx$.

Accepted Answer

$\sqrt{x^2+1} - \ln|x + \sqrt{x^2+1}| + C$

Question 19

Which of the following is the correct value of $\sin(\pi)$?

Accepted Answer

$0$

Calculus I-II Review - Functions, Derivatives, Integrals, and Trigonometric Identities

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