Question 1

Which of the following intervals is the domain where $$f(x) = \sqrt{x-5} is $$continuous?

Accepted Answer

$$[5, \infty)$$

Question 2

Given $$f(x) = \ln(x)$$, which of the following statements about its continuity is correct?

Accepted Answer

It is continuous for $$x \u003e 0.$$

Question 3

Consider the piecewise function $$f(x) = \begin{cases} 1, & x \u003c 1 \ 2, & x \geq 1 \end{cases}. $$What type of discontinuity does $$f(x)$$ have at $$x=1$$?

Accepted Answer

Jump discontinuity

Question 4

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

Accepted Answer

$$(-\infty, -1) \cup (-1, 1) \cup (1, \infty)$$

Question 5

Using the Intermediate Value Theorem, does $$f(x) = \sqrt{x}$$ have a root in $$[-1,1]$$?

Accepted Answer

No, because $$f(x) is $$not defined for $$x \u003c 0.$$

Question 6

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

Accepted Answer

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

Question 7

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 8

Find the average rate of change of $$f(x) = \ln(x)$$ over the interval $$[1,3].$$

Accepted Answer

$$\frac{\ln(3) - \ln(1)}{3-1}$$

Question 9

If $$V(t) = 6000 \left(1 - \frac{t}{60}\right)$$, what is the rate at which water is flowing out of the tank after $$30$$ minutes?

Accepted Answer

$$V'(30) = -100$$

Question 10

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

Accepted Answer

$$2x + 3$$

Question 11

For $$f(x) = \sin(x^2$ + 7)$$, which rule is used to differentiate $$f(x)$$?

Accepted Answer

Chain Rule

Question 12

Differentiate $$f(x) = \ln(1 + x^2$).$$

Accepted Answer

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

Question 13

The power that a battery supplies to an apparatus is $$P = \frac{V^2$R}{(R+r)^2$}$$, where $$r is $$the internal resistance. What happens to $$P as$$ $$R 	o 0$$?

Accepted Answer

$$P 	o 0$$

Question 14

Find $$\frac{d}{dx} \left( 	an$$^{-1}$$(\sqrt{x^2$-1}) ight).$$

Accepted Answer

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

Question 15

Which of the following functions is not differentiable at $$x=0$$?

Accepted Answer

$$f(x) = |x|$$

Question 16

If $$f(x) = \frac{e^x - e$$^{-x}$$}{2}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 17

Given $$f(x) = x^2$ + 4x + 3$$, what is the equation of the tangent line at $$x=0$$?

Accepted Answer

$$y = 4x + 3$$

Question 18

Which method should be used to differentiate $$f(x) = x^2$ e$$^{x^2}$$?

Accepted Answer

Product Rule

Question 19

The van der Waals equation for a mole of gas is $$\left(P + \frac{a}{V^2$}\right)(V-b) = nRT. $$What does the term $$\frac{a}{V^2$}$$ represent?

Accepted Answer

Correction for intermolecular attractions

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

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