Q: Given $$\lim_{x \to 0} f(x) = 0$$ and $$\lim_{x \to 0} g(x) = \infty$$, which of the following is true?

$$\lim_{x \to 0} \frac{f(x)}{g(x)} = 0$$

Question 1

The graph below shows the number of tuberculosis deaths in the United States from 1989 to 1998. Estimate the average rate of change in tuberculosis deaths from 1991 to 1997.

Accepted Answer

About $$-60$$ deaths per year

Question 2

Given $$\lim_{x 	o 0} f(x) = 0$$ and $$\lim_{x 	o 0} g(x) = \infty$$, which of the following is true?

Accepted Answer

$$\lim_{x 	o 0} \frac{f(x)}{g(x)} = 0$$

Question 3

Given that $$x \cdot \cos \left( \frac{1}{x} ight)$$ holds for all values of $$x \u003e 0$$, find $$\lim_{x 	o 0^+} x \cdot \cos \left( \frac{1}{x} ight).$$

Accepted Answer

$$0$$

Question 4

Find the value of $$k$$ that makes the function $$f(x) = \begin{cases} x^2$, & x \leq 9 \ k, & x \u003e 9 \end{cases}$$ everywhere continuous.

Accepted Answer

$$k = 81$$

Question 5

Evaluate $$\lim_{x 	o 2} \frac{x^2$ - 7x + 6}{x^2$ - 8x + 7}.$$

Accepted Answer

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

Question 6

A rocket in outer space is accelerating in a straight line. Its distance in meters from its starting point $$t$$ seconds after firing its thruster is $$s(t) = t^4$. Find the velocity of the rocket after 2$ seconds of the flight.

Accepted Answer

$$32 m/s$$

Question 7

The graphs below show the positions, velocity $$v = ds/dt$$, and acceleration $$a = d^2$s/dt^2$ of a body moving along a coordinate line as functions of time. Which graph is which?

Accepted Answer

A = position, B = velocity, C = acceleration

Question 8

Given the function $$f(x)$$ shown in the graph, what is $$f'(0)$$?

Accepted Answer

$$-1$$

Question 9

Consider the function $$f(x) = 4x^2$, x$$ \geq 0$$. Find the value of $$\frac{d}{dx}f(x)$$ at x = 2$.

Accepted Answer

$$16$$

Question 10

Let $$u$$ and $$v be $$differentiable functions of $$x$$ such that $$u(1) = 5$$, $$u'(1) = 6$$, $$v(1) = 7$$, $$v'(1) = -2. $$Find the value of $$\frac{d}{dx} \left( \frac{u}{v} \right) at$$ $$x = 1.$$

Accepted Answer

$$\frac{32}{49}$$

Question 11

Differentiate $$y = 	an$$^{-1}$$ \left( \frac{4x}{5} ight).$$

Accepted Answer

$$\frac{20}{16x^2$ + 25}$$

Question 12

Find $$\lim_{x 	o 2^-} f(x)$$ for $$f(x) = \begin{cases} -3x + 2, & x  2 \end{cases}.$$

Accepted Answer

$$-4$$

Question 13

Find $$f(2)$$ for the function $$f(x)$$ shown in the graph.

Accepted Answer

$$1$$

Question 14

Find $$\lim_{x 	o 0} f(x)$$ for the function $$f(x)$$ shown in the graph.

Accepted Answer

$$2$$

Question 15

Find $$\lim_{x 	o 2^+} f(x)$$ for $$f(x) = \begin{cases} -3x + 2, & x  2 \end{cases}.$$

Accepted Answer

$$4$$

Question 16

Use calculus to find the equations of all vertical and horizontal asymptotes of $$y = \frac{2x^2$ + 8}{2x^2$ + 3x - 2}.$$

Accepted Answer

Vertical: $$x = -2$$, $$x = \frac{1}{2}$$; Horizontal: $$y = 1$$

Question 17

Given $$c(t) = 114.9 e$$^{0.345t}$$, where t$ is years after 2007, which time interval has the greater average growth rate in smartphone sales: 2007–2009 or 2012–2014?

Accepted Answer

2012–2014

Question 18

Find the instantaneous growth rate in smartphone sales in 2008 and 2013 for $$c(t) = 114.9 e$$^{0.345t}$$. At which time was the instantaneous growth rate greater?

Accepted Answer

2013

Question 19

Use logarithmic differentiation to find $$\frac{dy}{dx}$$ for $$y = x$$^{3x}$$.

Accepted Answer

$$y' = x$$^{3x}$$ (3 \ln x + 3)$$

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