Question 1

Which of the following intervals is the domain of the function $$f(x) = \sqrt{7 - x}$$?

Accepted Answer

$$(-\infty, 7]$$

Question 2

For the function $$f(x) = \ln(|x|)$$, on which interval is $$f(x)$$ continuous?

Accepted Answer

$$(-\infty, 0) \cup (0, \infty)$$

Question 3

Given the piecewise function $$f(x) = \begin{cases} x^2$ + 2x, & x \u003c 2 \ 7 - x, & x \geq 2 \end{cases}$$, what value of the constant $$c$$ makes $$f(x)$$ continuous at $$x=2$$?

Accepted Answer

Question 4

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{3x^2$ + 4x}{5x^2$ + 6x}.$$

Accepted Answer

$$\frac{3}{5}$$

Question 5

The population of a culture of bacteria is given by $$p(t) = \frac{2500t}{t+1}$$, where $$t is in $$days. What happens to the population as $$t 	o \infty$$?

Accepted Answer

It approaches $$2500$$

Question 6

Let $$f(x) = \sqrt{x}. $$What is the average rate of change of $$f$$ over $$[1, 9]$$?

Accepted Answer

$$\frac{2}{3}$$

Question 7

Given $$f(x) = \ln(x)$$, what is the equation of the tangent line to $$f at$$ $$x = 1$$?

Accepted Answer

$$y = x - 1$$

Question 8

A cylindrical tank holds 6000 gallons of water. The volume $$V of $$water remaining in the tank after $$t$$ minutes is $$V(t) = 6000 \left(1 - \frac{t}{60}\$$right)^2$$. What is the rate at which water is flowing out of the tank after 30 minutes?

Accepted Answer

$$-200$$ gallons per minute

Question 9

For the function $$f(x) = \frac{1}{x^2$ - 2}$$, on which interval is $$f(x)$$ continuous?

Accepted Answer

$$(-\infty, -\sqrt{2}) \cup (-\sqrt{2}, \sqrt{2}) \cup (\sqrt{2}, \infty)$$

Question 10

Let $$f(x) = 	an$$^{-1}$$(\sqrt{x^2$ - 1}). $$What is $$\frac{d}{dx} f(x)$$?

Accepted Answer

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

Question 11

The function $$f(x) = \frac{e^x - 1}{e^x + 1}$$ has which of the following horizontal asymptotes?

Accepted Answer

$$y = 1$$

Question 12

Suppose the average blood alcohol concentration (BAC) after consumption of 15 mL of ethanol is modeled by $$C(t) = 0.0225e$$^{-0.086t}$$, where t$ is measured in minutes. How rapidly was the BAC increasing after 10 minutes?

Accepted Answer

$$-0.0171 mg/mL $$per minute

Question 13

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

Accepted Answer

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

Question 14

Let $$f(x) = x^2$ e^x. $$Which rule is required to differentiate $$f(x)$$?

Accepted Answer

Product Rule

Question 15

The equation in motion of an object is given by $$f(t) = 3t^2$ - 24t^2$ + 5. $$What is the acceleration of the object at $$t = 1$$ second?

Accepted Answer

$$-42 ft/s$^2$$

Question 16

Given $$y = x^2$ + y^2$ = 6$$, what is $$\frac{dy}{dx}$$ using implicit differentiation?

Accepted Answer

$$\frac{dy}{dx} = \frac{-x}{y}$$

Question 17

The power that a battery supplies to an apparatus is $$P = \frac{V^2$R}{(R + r)^2$}$$, where $$V$$ and $$r$$ are constants. What is $$\frac{dP}{dR}$$?

Accepted Answer

$$\frac{V^2$(r - R)}{(R + r)^3$}$$

Question 18

Which of the following statements about the Intermediate Value Theorem is correct?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$N is $$between $$f(a)$$ and $$f(b)$$, then there exists $$c \in (a, b)$$ such that $$f(c) = N.$$

Question 19

Let $$f(x) = \sin^2$(x^4 + 7$$)$$. What is $$\frac{df}{dx}$?

Accepted Answer

$$8x^3$$ \$$sin(x^4 + 7$$) \$$cos(x^4 + 7$$)$$

Calculus I Study Guide: Continuity, Limits, Derivatives, and Differentiation Rules

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