Q: Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

A skydiver jumps from a plane and his height above the ground at time $$t$$ seconds is given by $$h(t) = -16t^2$ + 2880. $$What is the average speed of the skydiver during the first minute of his jump ($$t = 0 to$$ $$t = 60$$)?

Accepted Answer

$$-32 ft/s$$

Question 2

Given $$h(t) = -16t^2$ + 2880$$, use the definition of the derivative to find the instantaneous speed of the skydiver at $$t = 1$$ second.

Accepted Answer

$$-32 ft/s$$

Question 3

Find the equation of the tangent line to the skydiver's position function $$h(t) = -16t^2$ + 2880 at $$the point $$(1, 2864).$$

Accepted Answer

$$y = -32(t - 1) + 2864$$

Question 4

Evaluate $$\lim_{x 	o \frac{\pi}{2}} (\sin(x) - 3\cos(x)).$$

Accepted Answer

$$-3$$

Question 5

Evaluate $$\lim_{x 	o 0} \frac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 6

Evaluate $$\lim_{x 	o 4} \frac{\sqrt{x+2} - 2}{x - 4}.$$

Accepted Answer

$$\frac{1}{4}$$

Question 7

Evaluate $$\lim_{x 	o 3} \frac{7x - 21}{x - 3}.$$

Accepted Answer

$$7$$

Question 8

Evaluate $$\lim_{x 	o 2} \frac{x^2$ + 1}{x^2$ - 5x + 6}.$$

Accepted Answer

$$\frac{5}{-4}$$

Question 9

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

Accepted Answer

$$-1$$

Question 10

For the function $$f(x) = x^3$ - 5x^2$ - 10x^2$ + 15x + 30$$, does the equation $$f(x) = 0$$ have a solution in the interval $$[2, 3]$$? Explain using the Intermediate Value Theorem.

Accepted Answer

Yes, because $$f(2)$$ and $$f(3)$$ have opposite signs.

Question 11

Describe the set of $$x-$$values where the function $$f(x) = \frac{x^2$ - 1}{x^2$ - 4} is $$continuous using interval notation.

Accepted Answer

$$(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$$

Question 12

For the function $$f(x) = \frac{x^2$ - 1}{x^2$ - 4}$$, which points of discontinuity are removable?

Accepted Answer

None; both $$x = 2$$ and $$x = -2$$ are non-removable discontinuities.

Question 13

Evaluate $$\lim_{x 	o 2} (f(x) + 3g(x)) if$$ $$f(x)$$ and $$g(x)$$ are continuous at $$x = 2$$ and $$f(2) = 5$$, $$g(2) = -1.$$

Accepted Answer

$$8$$

Question 14

On the interval $$[-1, 1]$$, the inequality $$-1 \leq \sin\left(\frac{\pi}{2}xight) \leq 1 is $$true everywhere except at $$x = 0. $$Use this fact to evaluate $$\lim_{x 	o 0} (\sin(\cos(\frac{\pi}{2}x))).$$

Accepted Answer

$$\sin(1)$$

Question 15

A chemical reaction's rate is modeled by $$r(t) = 5e$$^{-0.2t}$$. What is the instantaneous rate of change at t = 10$ seconds?

Accepted Answer

$$-1e^{-2}$$

Question 16

A patient’s temperature $$T(t) is $$modeled by $$T(t) = 98.6 + 0.5e$$^{-0.1t}$$, where t$ is hours after medication. What is the temperature after 5 hours?

Accepted Answer

$$98.6 + 0.5e$$^{-0.5}$$

Question 17

Which of the following best describes the meaning of the derivative $$f'(a) at$$ $$x = a$$?

Accepted Answer

The slope of the tangent line to $$f(x) at$$ $$x = a$$

Question 18

Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Calculus I Exam Study Guide: Limits, Continuity, and Tangents

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