Q: Consider $$f(x) = \frac{x^2$-1}{x-1}. Is $$the discontinuity at $$x=1$$ removable? Why?

Yes, because $$\lim_{x \to 1} f(x)$$ exists and equals $$2$$

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 during the first minute after the jump?

Accepted Answer

$$-960 ft/sec$$

Question 2

Given $$h(t) = -16t^2$ + 2880$$, what is the instantaneous velocity of the skydiver at $$t = 1$$ second?

Accepted Answer

$$-64 ft/sec$$

Question 3

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

Accepted Answer

$$y = -64(t-1) + 2864$$

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 $$\lim_{x 	o 4} \frac{\sqrt{x+5} - 3}{x-4}.$$

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

$$-2$$

Question 10

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

Accepted Answer

$$(-\infty, 1) \cup (1, \infty)$$, but the discontinuity at $$x=1 is $$removable

Question 11

Consider $$f(x) = \frac{x^2$-1}{x-1}. Is $$the discontinuity at $$x=1$$ removable? Why?

Accepted Answer

Yes, because $$\lim_{x 	o 1} f(x)$$ exists and equals $$2$$

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

Evaluate $$\lim_{x 	o 0} (|\cos(\frac{x}{2})|)$$ and explain using the squeeze theorem.

Accepted Answer

$$1$$, because $$|\cos(\frac{x}{2})| is $$squeezed between $$-1$$ and $$1$$ and is continuous at $$x=0$$

Question 14

A chemical reaction's temperature in degrees Celsius is modeled by $$T(t) = 100e$$^{-0.05t}$$, where t$ is time in minutes. What is the instantaneous rate of change of temperature at t=0$?

Accepted Answer

$$-5$$ degrees/min

Question 15

A patient’s blood concentration of a drug is given by $$C(t) = 50 - 2t^2$, where t$ is hours after administration. When does the concentration reach zero?

Accepted Answer

$$t = 5$$ hours

Question 16

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b). $$What theorem guarantees that $$f(x) = 0$$ has a solution in $$(a, b)$$?

Accepted Answer

Intermediate Value Theorem

Question 17

A ball is thrown upward from a height of $$80$$ meters with an initial velocity of $$20 m/s. $$Its height is $$h(t) = -5t^2$ + 20t + 80. $$What is the maximum height reached by the ball?

Accepted Answer

Question 18

Evaluate $$\lim_{x 	o 0} \frac{e$$^{2x}$$ - 1}{x}.$$

Accepted Answer

$$2$$

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

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