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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