Q: Which of the following is a necessary condition for applying L'Hospital's Rule to $$\lim_{x \to c} \frac{f(x)}{g(x)}$$?

$$\lim_{x \to c} f(x) = 0$$ and $$\lim_{x \to c} g(x) = 0$$

Question 1

Given the graph of $$f(x) ($$see image 1), on which interval is the function $$f(x)$$ constant?

Accepted Answer

$$(-2, 3)$$

Question 2

For the function $$f(x) in $$image 1, at which $$x-$$value does $$f(x)$$ attain a local maximum?

Accepted Answer

$$x = -4$$

Question 3

Suppose $$f'(x) is $$given by the graph in image 2. On which interval is $$f(x)$$ concave downward?

Accepted Answer

$$(-2, 2)$$

Question 4

According to the table in image 3, if $$f''(x) \u003c 0$$ and $$f'(x) is $$decreasing, what can you conclude about the graph of $$f(x)$$?

Accepted Answer

It is concave downward.

Question 5

Given the table in image 5, if $$f'(c) = 0$$ and $$f''(c) \u003e 0$$, what does this indicate about $$f(x) at$$ $$x = c$$?

Accepted Answer

There is a local minimum at $$x = c.$$

Question 6

If $$f'(x) \u003e 0$$ for all $$x in an $$interval, what can you say about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$increasing.

Question 7

A company’s sales function is $$N(x) = 3x^3$ - 0.25x^4$ + 200$$ for $$0 \leq x \leq 9$$, where $$x is $$thousands of dollars spent on advertising. At what value of $$x$$ does the point of diminishing returns occur?

Accepted Answer

$$x$$ where $$N''(x) = 0$$

Question 8

Evaluate $$\lim_{x 	o 1} \frac{e^x - e}{x - 1}$$ using L'Hospital's Rule.

Accepted Answer

$$e$$

Question 9

Which of the following is a necessary condition for applying L'Hospital's Rule to $$\lim_{x 	o c} \frac{f(x)}{g(x)}$$?

Accepted Answer

$$\lim_{x 	o c} f(x) = 0$$ and $$\lim_{x 	o c} g(x) = 0$$

Question 10

Suppose $$f(x) is $$continuous on $$[a, b]. $$According to the Extreme Value Theorem, where must the absolute maximum and minimum of $$f(x)$$ occur?

Accepted Answer

At critical numbers or endpoints.

Question 11

Given $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, find the $$x-$$values where $$f(x)$$ has inflection points.

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 12

A ball is launched upward from a 112 ft building with velocity 96 ft/s: $$s(t) = -16t^2$ + 96t + 112. $$When does the ball reach its maximum height?

Accepted Answer

$$t = 3$$ seconds

Question 13

If $$f''(x) \u003e 0$$ for all $$x in an $$interval, what does this say about the graph of $$f(x)$$?

Accepted Answer

It is concave upward.

Question 14

Given $$f(x) = x^2$ e^x$$, what is $$f''(x)$$?

Accepted Answer

$$x^2 e^x + 4x e^x + 2e^x$

Question 15

A rectangular garden is to be fenced with $$320$$ dollars. Three sides use wire at $$2/ft$$, one side uses wood at $$6/ft. $$What is the optimal length of the wood side for maximum area?

Accepted Answer

$$40 ft$$

Chapter 4: Graphing and Optimization – Business Calculus Study Notes

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