Q: Given $$h(x) = 0.001x^3$ - 0.24x^2$ + 14.4x$$, which of the following is the correct expression for the velocity of the hot air balloon at $$x=80$$ minutes?

$$h'(80) = 0.003 \times 80^2$ - 0.48 \times 80 + 14.4$$

Q: Which of the following is the correct limit definition of the derivative for $$f(x) = 6x^2$ - 8x - 6$$?

$$f'(x) = \lim\limits_{h \to 0} \dfrac{6(x+h)^2$ - 8(x+h) - 6 - [6x^2$ - 8x - 6]}{h}$$

Question 1

Given the graph above, which of the following statements about its asymptotes is correct?

Accepted Answer

It has a horizontal asymptote at $$y=0$$ and a vertical asymptote at $$x=4.$$

Question 2

Evaluate $$\lim\limits_{x 	o 0} f(x)$$ using the graph provided.

Accepted Answer

$$0$$

Question 3

For the function $$g(x) = -4x^2$ + 20x - 15$$, what is $$g'(x)$$?

Accepted Answer

$$-8x + 20$$

Question 4

Given $$h(x) = \sqrt{x^2$ + 8}$$, what is $$h'(x)$$?

Accepted Answer

$$\dfrac{x}{\sqrt{x^2$ + 8}}$$

Question 5

A pool is being drained for the winter. The amount of water left in the pool (in gallons) is given by $$g(x) = 15x^2$ - 50x + 4000$$, where $$x is $$the number of minutes since draining began. What is the average rate of change of water volume between 50 and 100 minutes?

Accepted Answer

$$-27.5$$ gal/min

Question 6

For the same pool draining scenario, what is the instantaneous rate at which water is draining exactly 80 minutes after draining began?

Accepted Answer

$$-26$$ gal/min

Question 7

Given $$h(x) = 0.001x^3$ - 0.24x^2$ + 14.4x$$, which of the following is the correct expression for the velocity of the hot air balloon at $$x=80$$ minutes?

Accepted Answer

$$h'(80) = 0.003 	imes 80^2$ - 0.48 	imes 80 + 14.4$$

Question 8

For the function $$k(x) = 3x^2$ - 4x^4$, what is k$$''(x)$$, the second derivative?

Accepted Answer

$$6 - 48x^2$

Question 9

Find all points of non-differentiability for the function shown in the graph.

Accepted Answer

$$x = -4, -3, 1, 2, 4$$

Question 10

Is the function shown in the graph continuous over the interval $$(-2, 2)$$?

Accepted Answer

Yes, it is continuous over $$(-2, 2).$$

Question 11

Given $$f(x) = 2x^3$ - x - 5$$, use the limit definition of the derivative to find $$f'(x).$$

Accepted Answer

$$6x^2 - 1$

Question 12

Find the equation of the tangent line to $$f(x) = 4x^2$ - 2x + 10 at$$ $$x=3.$$

Accepted Answer

$$y = 22x - 31$$

Question 13

For $$g(x) = x^2$ - 2x^5$ + 12$$, what is the equation of the tangent line at $$x=3$$?

Accepted Answer

$$y = -49x + 96$$

Question 14

For $$h(x) = 7x^2$ - 75x + 12$$, at what value of $$x$$ does the tangent line have a slope of $$0$$?

Accepted Answer

$$x = 5.5$$

Question 15

Given $$h(x) = 6x^3$ - 36x^2$ + 40$$, at what value of $$x$$ does the tangent line have a slope of $$0$$?

Accepted Answer

$$x = 6.2$$

Question 16

For $$g(x) = \dfrac{2x^2$ - 12x - 2}{x^4$}$$, what is $$g'(x)$$?

Accepted Answer

$$\dfrac{4x^3$ - 24x^2$ - 8x}{x^8$}$$

Question 17

Find the second derivative of $$f(x) = 24x^3$.

Accepted Answer

$$144x$$

Question 18

A hot air balloon's height is given by $$h(x) = 0.001x^3$ - 0.24x^2$ + 14.4x. Is $$the balloon rising or descending at $$x=80$$ minutes?

Accepted Answer

Descending

Question 19

Which of the following is the correct limit definition of the derivative for $$f(x) = 6x^2$ - 8x - 6$$?

Accepted Answer

$$f'(x) = \lim\limits_{h 	o 0} \dfrac{6(x+h)^2$ - 8(x+h) - 6 - [6x^2$ - 8x - 6]}{h}$$

Business Calculus Test 1 Review: Limits, Derivatives, Tangents, and Applications

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