Question 1

A skydiver's height above the ground is modeled by $$h(t) = -16t^2$ + 2880$$, where $$t is $$the time in seconds after the jump. What is the average speed of the skydiver between $$t = 0$$ and $$t = 60$$ seconds?

Accepted Answer

$$-48 ft/s$$

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

What is the equation of the tangent line to $$h(t) = -16t^2$ + 2880 at $$the point $$(1, 2864)$$?

Accepted Answer

$$y = -32x + 2896$$

Question 4

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

Accepted Answer

$$1 - 3 \cdot 0 = 1$$

Question 5

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

Accepted Answer

$$5$$

Question 6

Evaluate $$\lim\limits_{x 	o 4} \frac{\sqrt{x+5} - 3}{x - 4}.$$

Accepted Answer

$$\frac{1}{6}$$

Question 7

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

Accepted Answer

$$7$$

Question 8

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

Accepted Answer

$$\frac{5}{-2}$$

Question 9

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

Accepted Answer

$$-1$$

Question 10

For which values of $$x is $$the function $$f(x) = \frac{x^2$ + 1}{x^2$ - 5x + 6}$$ continuous?

Accepted Answer

$$(-\infty, 2) \cup (2, 3) \cup (3, \infty)$$

Question 11

At which $$x-$$values does $$f(x) = \frac{x^2$ + 1}{x^2$ - 5x + 6}$$ have a removable discontinuity?

Accepted Answer

Neither $$x = 2$$ nor $$x = 3$$

Question 12

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

Accepted Answer

Yes, by the Intermediate Value Theorem

Question 13

Given $$\lim\limits_{x 	o 0} \frac{\sin(x)}{x} = 1$$, use this fact to evaluate $$\lim\limits_{x 	o 0} \frac{\sin(2x)}{x}.$$

Accepted Answer

$$2$$

Question 14

Suppose $$f(x) is $$continuous everywhere except at $$x = 2$$ and $$x = 3. At$$ $$x = 2$$, $$\lim\limits_{x 	o 2} f(x)$$ exists and equals $$5$$, but $$f(2) = 7. $$What type of discontinuity is this?

Accepted Answer

Removable discontinuity

Question 15

Evaluate $$\lim\limits_{x 	o 0} \left| \sin \left( \cos \left( \frac{x}{2} ight) ight) ight|.$$

Accepted Answer

$$|\sin(1)|$$

Question 16

A patient’s blood concentration of a drug is modeled by $$C(t) = 10e$$^{-0.2t}$$, where t$ is in hours. What is the instantaneous rate of change of C$ at t = 5$ hours?

Accepted Answer

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

Question 17

A chemical reaction’s temperature is given by $$T(t) = 100 - 5t^2$. What is the average rate of change of temperature from t = 2$ to t = 4$?

Accepted Answer

$$-30$$

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

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