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

$$-960 ft/$$min$

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

$$y = -32x + 2896$$

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 $$\lim_{x 	o 3} \frac{7x - 21}{x^2$ - 9}.$$

Accepted Answer

$$\frac{7}{6}$$

Question 8

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

Accepted Answer

$$\frac{5}{12}$$

Question 9

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

Accepted Answer

$$\frac{3}{\sqrt{10}}$$

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 $$\lim_{x 	o 2} [f(x) + 3g(x)]$$ given that $$\lim_{x 	o 2} f(x) = 5$$ and $$\lim_{x 	o 2} g(x) = -1.$$

Accepted Answer

$$8$$

Question 14

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

Accepted Answer

$$1$$

Question 15

A chemical reaction's rate is modeled by $$r(t) = 5t^2$ - 3t + 2. $$What is the instantaneous rate of reaction at $$t = 4$$ seconds?

Accepted Answer

$$38$$

Question 16

A medication's concentration in the bloodstream is given by $$C(t) = 100e$$^{-0.2t}$$. What is the rate of change of concentration at t = 5$ hours?

Accepted Answer

$$-20e^{-1}$$

Question 17

A car's position is given by $$s(t) = 2t^3$ - 5t^2$ + 4t. $$What is the car's acceleration at $$t = 2$$ seconds?

Accepted Answer

$$16$$

Question 18

Which of the following statements about the limit $$\lim_{x 	o a} f(x) is $$true?

Accepted Answer

The limit exists if the left and right limits are equal.

Calculus I: Limits, Continuity, and Tangent Lines – Study Notes

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