Q: Find the antiderivative of $$f(x) = 2\sqrt{x} + 6\sec^2$(3x).$$

$$\frac{4}{3}x$$^{3/2}$$ + 2\tan(3x) + C$$

Question 1

Given the function $$f(x) = x^4$ + 4x^3$, for which intervals is f(x$$)$$ concave up?

Accepted Answer

$$x \u003e 0$$

Question 2

For the function $$f(x) = x^4$ + 4x^3$, at which x$-value does an inflection point occur?

Accepted Answer

$$x = -2$$

Question 3

A hot air balloon rises vertically at a constant rate of $$4 m/s. An $$observer is $$200$$ meters from the launch site. How fast is the angle of elevation (in radians per second) increasing $$30$$ seconds after launch?

Accepted Answer

$$0.016$$

Question 4

Find the equation of the tangent line to the curve $$4x^3$$ = $$y^2(4$$ - $$y^2$$)$$ at the point where x = 2$.

Accepted Answer

$$y = 8x - 16$$

Question 5

A rectangular bird sanctuary is to be built with one side along a river (no fence needed there) and $$12 km of $$fencing for the other three sides. What are the dimensions that maximize the area?

Accepted Answer

$$3 km by$$ $$6 km$$

Question 6

A rectangular bird sanctuary with one side along a river is to have an area of $$8 km$^2. $$What dimensions minimize the amount of fencing required for the other three sides?

Accepted Answer

$$4 km by$$ $$2 km$$

Question 7

Evaluate the integral $$\int (3x^3$ + 1) dx.$$

Accepted Answer

$$\frac{3}{4}x^4$ + x + C$$

Question 8

Find the antiderivative of $$f(x) = 2\sqrt{x} + 6\sec^2$(3x).$$

Accepted Answer

$$\frac{4}{3}x$$^{3/2}$$ + 2	an(3x) + C$$

Question 9

For the function $$f(x) = x^4$ + 4x^3$, at which x$-value does a local minimum occur?

Accepted Answer

$$x = -3$$

Question 10

Given $$f(x) = x^4$ + 4x^3$, on which interval is f(x$$)$$ decreasing?

Accepted Answer

$$x \u003c -3$$

Question 11

Evaluate the integral $$\int 2\sec^2$(3x) dx.$$

Accepted Answer

$$\frac{2}{3}	an(3x) + C$$

Question 12

Find the antiderivative of $$f(x) = \cos(2x).$$

Accepted Answer

$$\frac{1}{2}\sin(2x) + C$$

Question 13

Evaluate the integral $$\int (2x + 1)^2$ dx.$$

Accepted Answer

$$\frac{4}{3}x^3$ + 4x^2$ + x + C$$

Question 14

If $$f(x) = x^4$ + 4x^3$, what is f$$''(x)$$?

Accepted Answer

$$12x^2 + 24x$

Question 15

A tangent line to a curve at $$x = 2$$ has a slope of $$8. $$What is the slope of the normal line at that point?

Accepted Answer

$$-\frac{1}{8}$$

Question 16

Suppose a function $$f(x)$$ has $$f'(x) = 0 at$$ $$x = a$$ and $$f''(a) \u003e 0. $$What does this indicate about $$f(x) at$$ $$x = a$$?

Accepted Answer

There is a local minimum at $$x = a$$

Question 17

A rectangular enclosure is to be built along a river, using $$12 km of $$fencing for the other three sides. If $$x is $$the length parallel to the river, what is the area as a function of $$x$$?

Accepted Answer

$$A(x) = x(12 - 2x)$$

Question 18

Given $$\int (x - 5) dx$$, what is the result?

Accepted Answer

$$\frac{1}{2}x^2$ - 5x + C$$

Calculus Study Guide: Derivatives, Applications, and Integrals

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