Question 1

A company’s profit function is given by $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold. What is the critical value of $$x$$?

Accepted Answer

$$x = 2$$

Question 2

Given $$f(x) = x^3$ - 3x^2$ + 2$$, use the first derivative test to determine where $$f(x) is $$increasing.

Accepted Answer

$$x  2$$

Question 3

For $$f(x) = x^4$ - 4x^2$, use the second derivative test to classify the critical point at x = 0$.

Accepted Answer

Saddle point

Question 4

A hiker’s elevation is modeled by $$E(t) = -t^2$ + 6t + 5$$ for $$t in$$ $$[0, 8]$$ hours. What is the absolute maximum elevation?

Accepted Answer

$$E(3) = 14$$

Question 5

Given $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, on which interval is $$f(x)$$ decreasing?

Accepted Answer

$$1 \u003c x \u003c 3$$

Question 6

For $$f(x) = x^4$ - 4x^2$, where is the function concave up?

Accepted Answer

$$x  1$$

Question 7

A farmer wants to maximize the area of a rectangular field with 400 meters of fencing. What is the maximum area using derivatives?

Accepted Answer

$$25,000 m$^2$$

Question 8

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x}$$ using L'Hôpital's Rule.

Accepted Answer

$$1$$

Question 9

Find an antiderivative of $$f(x) = 3x^2$.

Accepted Answer

$$x^3 + C$

Question 10

Given $$F'(x) = 2x$$ and $$F(1) = 5$$, find the integration constant $$C.$$

Accepted Answer

$$C = 4$$

Question 11

Evaluate $$\$$int_0$$^2 x^2$ dx$$ using the definition of the Riemann sum.

Accepted Answer

$$\frac{8}{3}$$

Question 12

Use the Fundamental Theorem of Calculus to find the area under $$f(x) = 2x$$ from $$x = 1 to$$ $$x = 4.$$

Accepted Answer

$$15$$

Question 13

If $$F(x) = \$$int_0$$^x \cos t dt$$, what is $$F'(x)$$?

Accepted Answer

$$\cos x$$

Question 14

Evaluate $$\int (2x+1)^5$ dx$$ using substitution.

Accepted Answer

$$\frac{1}{12}(2x+1)^6$ + C$$

Question 15

Evaluate $$\$$int_0$$^1 2x e$$^{x^2}$$ dx$$ using substitution.

Accepted Answer

$$e - 1$$

Question 16

A tank is being filled so that the rate of water flow is $$r(t) = 3t^2$ liters per minute. How much water enters the tank from t = 0$ to t = 2$ minutes?

Accepted Answer

$$8$$ liters

Question 17

Find the area between $$y = x^2$ and y = x$ from x = 0$ to x = 1$.

Accepted Answer

$$\frac{1}{6}$$

Calculus Study Guide: Derivatives, Integrals, and Applications

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