Question 1

Which of the following best describes the geometric meaning of the derivative $$f'(x) at a $$point $$x=a$$?

Accepted Answer

It gives the slope of the tangent line to $$f(x) at$$ $$x=a.$$

Question 2

Given $$f(x) = x^3$ \sin(x)$$, which rule must be used to find $$f'(x)$$?

Accepted Answer

Product rule

Question 3

Evaluate $$\displaystyle \int 2x \cos(x^2$) \, dx$$ using $$u-$$substitution.

Accepted Answer

$$\sin(x^2$) + C$$

Question 4

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

Accepted Answer

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

Question 5

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$F(x) is an $$antiderivative of $$f(x). $$According to the Fundamental Theorem of Calculus, $$\int_a^b f(x)\,dx$$ equals:

Accepted Answer

$$F(b) - F(a)$$

Question 6

Find the area between the curves $$f(x) = x^2$ and g(x) = x$ from x=0$ to x=1$.

Accepted Answer

$$\displaystyle \$$int_0$$^1 (x - x^2$)\,dx$$

Question 7

Which of the following is the correct $$u-$$substitution for $$\int x e$$^{x^2}$$ dx$$?

Accepted Answer

Let $$u = x^2$, then du = 2x dx$

Question 8

If $$y = \sin(x^2$)$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$2x \cos(x^2$)$$

Question 9

A medication’s concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$. What is the instantaneous rate of change of concentration at t=0$?

Accepted Answer

$$-1$$

Question 10

Which of the following integrals represents the total area between $$y = \cos(x)$$ and $$y = \sin(x)$$ from $$x=0 to$$ $$x=\pi$$?

Accepted Answer

$$\displaystyle \$$int_0$$^\pi |\cos(x) - \sin(x)| dx$$

Question 11

Which rule is used to differentiate $$y = 	an$$^{-1}$$(x)$$?

Accepted Answer

Inverse trigonometric rule

Question 12

Evaluate $$\displaystyle \int \dfrac{1}{x^2$ + 1} dx.$$

Accepted Answer

$$	an$$^{-1}$$(x) + C$$

Question 13

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

Accepted Answer

$$2e^{2x}$$

Question 14

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

Accepted Answer

$$\dfrac{1}{x}$$

Question 15

Which of the following is the correct formula for the area between two curves $$f(x)$$ and $$g(x)$$ from $$x=a to$$ $$x=b$$?

Accepted Answer

$$\displaystyle \int_a^b [f(x) - g(x)] dx$$

Question 16

Evaluate $$\displaystyle \int \cosh(x) dx.$$

Accepted Answer

$$\sinh(x) + C$$

Calculus I & II Review: Differentiation, Integration, and Area Between Curves

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