Question 1

Evaluate the following limit: $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 2

Which of the following statements about the derivative of a constant function $$f(x) = c is $$true?

Accepted Answer

The derivative is $$0$$

Question 3

A ball is thrown vertically upward with a height function $$h(t) = 20t - 5t^2$. What is the velocity of the ball at t = 2$ seconds?

Accepted Answer

$$-10 m/s$$

Question 4

Find $$\frac{dy}{dx} if$$ $$y = (3x^2$ + 2x)(x^3$ - 1).$$

Accepted Answer

$$9x^4$$ + $$6x^3$$ + $$6x^2 - 2x$

Question 5

If $$f(x) = \sin x$$, what is $$\lim_{x 	o 0} \frac{\sin x}{x}$$?

Accepted Answer

$$1$$

Question 6

Differentiate $$y = e$$^{2x}$$ \cdot \ln x$$ with respect to $$x.$$

Accepted Answer

$$2e^{2x}$$ \ln x + \frac{$$e^{2x}$$}{x}$$

Question 7

A company’s profit $$P (in $$thousands) is given by $$P(x) = -2x^2$ + 12x - 16$$, where $$x is $$the number of units sold (in thousands). For what value of $$x is $$the profit maximized?

Accepted Answer

$$x = 4$$

Question 8

Find the equation of the tangent to the curve $$y = x^3$ - 3x^2$ + 2 at$$ $$x = 2.$$

Accepted Answer

$$y = -1x + 2$$

Question 9

If $$y = 	an$$^{-1}$$(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 10

Given $$f(x) = x^4$ - 4x^3$ + 6x^2$, find all values of x$ where f$$''(x) = 0$$.

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 11

Differentiate $$y = \ln(3x^2$ - 5)$$ with respect to $$x.$$

Accepted Answer

$$\frac{6x}{3x^2$ - 5}$$

Question 12

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

Accepted Answer

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

Question 13

A tank is being filled so that the volume of water at time $$t is$$ $$V(t) = 5t^2$ + 2t. $$What is the rate at which water is being added at $$t = 3$$ minutes?

Accepted Answer

$$32$$ units/min

Question 14

Find $$\frac{dy}{dx} if$$ $$y = \frac{5x - 2}{x^2$ + 1}.$$

Accepted Answer

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

Question 15

If $$f(x) = x^2$ + 3x$$, use the definition of the derivative to find $$f'(1).$$

Accepted Answer

$$5$$

Question 16

Which of the following is the correct application of the chain rule for $$y = (2x + 1)^5$?

Accepted Answer

$$5(2x + 1)^4$ \cdot 2$$

Question 17

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

Accepted Answer

$$-3.68$$ units/hour

Question 18

Find the second derivative $$\frac{d^2$y}{dx^2$}$$ for $$y = x^3$ - 6x^2$ + 9x.$$

Accepted Answer

$$6x - 12$$

Question 19

If $$y = \sec^2$ x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2\sec^2$ x 	an x$$

Comprehensive Study Notes: Calculus Fundamentals and Applications

Study Guide - Practice Questions

Study Guide - Flashcards