Question 1

A fitness tracker records the distance (in meters) a runner covers at different times during a training session. The data is:

$$\begin{array}{cc}
	ext{Time (min)} & 	ext{Distance (m)} \
0 & 0 \
10 & 1200 \
20 & 2700 \
30 & 4500 \
40 & 6800 
\end{array}$$

Estimate the runner's speed at 20 minutes using the secant line between $$t = 10$$ and $$t = 30$$ minutes.

Accepted Answer

$$\frac{4500 - 1200}{30 - 10} = 165 m/$$min

Question 2

The amount of water drained from a tank over time is recorded as follows:

$$\begin{array}{cc}
	ext{Time (min)} & 	ext{Water Drained (L)} \
0 & 0 \
1 & 12.5 \
3 & 28.6 \
4 & 47.2 \
5 & 70.3 
\end{array}$$

Find the average rate of change of water drained from $$t = 1 to$$ $$t = 4$$ minutes.

Accepted Answer

Question 3

Given the function $$f(x) = x^2$ + 3x$$, what is the slope of the tangent line at the point $$(-1, -2)$$?

Accepted Answer

$$1$$

Question 4

For the function $$f(x) = \frac{3x + 1}{x^2$ - 4}$$, find all vertical asymptotes.

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 5

Evaluate $$\lim_{x 	o 0} \sqrt{x^3$ + x^2$} \sin \frac{1}{x}.$$

Accepted Answer

$$0$$

Question 6

Find the derivative of $$f(x) = x^3$ - 2x + 1$$ using the definition of the derivative.

Accepted Answer

$$3x^2 - 2$

Question 7

Suppose $$g$$ and $$f$$ are continuous at $$x = 4. If$$ $$f(4) = 3$$ and $$\lim_{x 	o 4} 5f(x) - g(x) = 12$$, find $$g(4).$$

Accepted Answer

$$3$$

Question 8

Find the equation of the tangent line to the curve $$y = 3x^2$ - 2x at $$the point $$(2, 8).$$

Accepted Answer

Question 9

For the function $$f(x) = \frac{x^2$ + 4}{x^2$ - 4}$$, what is the vertical asymptote?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 10

Evaluate $$\lim_{x 	o \infty} \frac{3x^2$ + 2x}{x^2$ - 4x + 1}.$$

Accepted Answer

$$3$$

Question 11

Find the derivative of $$g(x) = \sqrt{2x - 3}.$$

Accepted Answer

Question 12

Suppose $$\lim_{x 	o 2} f(x) = 5$$ and $$\lim_{x 	o 2} g(x) = 3. $$What is $$\lim_{x 	o 2} [f(x)g(x)]$$?

Accepted Answer

$$15$$

Question 13

Find the domain of $$g(x) = \frac{1}{x^2$ - 4}.$$

Accepted Answer

$$x 
eq 2$$ and $$x 
eq -2$$

Question 14

Find the value of $$c if$$ $$f(x) = \begin{cases} cx + 7, & x \leq 2 \ x^2$ - 5, & x \u003e 2 \end{cases} is $$continuous at $$x = 2.$$

Accepted Answer

$$c = 6$$

Question 15

Find the equation of the tangent line to $$y = x^3$ - 2x + 1 at$$ $$x = 2.$$

Accepted Answer

$$y = 10x - 11$$

Question 16

Use the Intermediate Value Theorem to show that $$f(x) = x^2$ + 2x - 8$$ has a root in the interval $$[1, 5].$$

Accepted Answer

Because $$f(1) = -5$$ and $$f(5) = 27$$, and $$0 is $$between $$-5$$ and $$27$$, there is a root in $$[1, 5].$$

Question 17

Find all asymptotes and holes of $$f(x) = \frac{x^2$ - 3x - 4}{x^2$ - x}.$$

Accepted Answer

Vertical asymptotes at $$x = 0$$ and $$x = 1$$, hole at $$x = -4$$

Question 18

Evaluate $$\lim_{x 	o 5} \frac{x + 2}{x - 6}.$$

Accepted Answer

$$-1$$

Question 19

Given $$f(x) = x^3$ + 2x + 1$$, find $$f'(x).$$

Accepted Answer

$$3x^2 + 2$

Question 20

If $$f(x) = x^2$ + 3x$$, what is the equation of the tangent line at $$x = -1$$?

Accepted Answer

$$y = x - 1$$

Calculus I Study Guide: Limits, Derivatives, Tangent Lines, and Asymptotes

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