Question 1

Decide whether the limit $$\lim_{x 	o 1^+} f(x)$$ exists for the function shown in the first image, and if it exists, find its value.

Accepted Answer

$$4$$

Question 2

For the function shown in the second image, find $$\lim_{x 	o -1^-} f(x)$$ and $$\lim_{x 	o -1^+} f(x).$$

Accepted Answer

$$-7, -2$$

Question 3

Let $$\lim_{x 	o 9} f(x) = 3$$ and $$\lim_{x 	o 9} g(x) = -1. $$Find $$\lim_{x 	o 9} \frac{[f(x)]^2}{6 + g(x)}.$$

Accepted Answer

$$\frac{9}{5}$$

Question 4

Use the properties of limits to decide whether the limit $$\lim_{x 	o \infty} \frac{6x^5$ - x + 4}{9x^2$ - x - 6}$$ exists. If it exists, find its value.

Accepted Answer

Does not exist

Question 5

Find all points where the function $$f(x) = \frac{x^2$ - 4}{x(x-2)} is $$discontinuous.

Accepted Answer

$$x = 0, x = 2$$

Question 6

Estimate the slope of the tangent line to the curve at the point $$(-4, 3) as $$shown in the third image.

Accepted Answer

$$-\frac{1}{2}$$

Question 7

Find the $$x-$$values where the function shown in the fourth image does not have a derivative.

Accepted Answer

$$x = -2, x = 0, x = 2$$

Question 8

Suppose the demand for a certain item is given by $$D(p) = -3p^2$ + 7p + 3$$, where $$p$$ represents the price of the item. Find $$D'(7).$$

Accepted Answer

$$-32$$

Question 9

Given the graph in the fifth image, which of the following graphs best represents its derivative?

Accepted Answer

Eighth image

Question 10

Find the derivative of $$y = (x^2$ + 4)^3$.

Accepted Answer

$$6x^5$$ + $$24x^3 + 48x$

Question 11

Find the slope of the line tangent to the graph of $$y = \frac{7}{x} - x at$$ $$x = 4.$$

Accepted Answer

$$-\frac{11}{16}$$

Question 12

The profit in dollars from the sale of $$x$$ thousand compact disc players is $$P(x) = x^3$ - 9x^2$ + 12x + 8. $$Find the marginal profit when $$x = 10.$$

Accepted Answer

$$190$$

Question 13

If the price of a product is given by $$P(x) = \frac{1024}{x} + 900$$, where $$x$$ represents the demand for the product, find the rate of change of price when the demand is $$2.$$

Accepted Answer

$$-512$$

Question 14

Use the product rule to find the derivative of $$f(x) = (3x^3$ + 3)(2x^7$ - 8). $$

Accepted Answer

$$60x^9$$ + $$42x^6 - 72x$

Question 15

Use the quotient rule to find the derivative of $$g(x) = \frac{x^2$ + 5}{x^2$ + 6x}.$$

Accepted Answer

$$\frac{6x^2$ - 10x - 30}{x^2$(x + 6)^2$}$$

Question 16

The total profit from selling $$x$$ units of cookbooks is $$P(x) = (7x - 5)(3x - 4). $$Find the marginal average profit function.

Accepted Answer

$$21 - \frac{43}{x^2$}$$

Question 17

Find $$f[g(x)]$$ and $$g[f(x)]$$ for $$f(x) = \frac{2}{x}$$ and $$g(x) = 2x^3$.

Accepted Answer

$$f[g(x)] = \frac{1}{x^3$};\ g[f(x)] = \frac{4}{x^3$}$$

Question 18

Write the function $$y = \frac{2}{3x + 2} as $$the composition of two functions $$f$$ and $$g$$ such that $$y = f[g(x)].$$

Accepted Answer

$$f(x) = \frac{2}{x},\ g(x) = 3x + 2$$

Business Calculus Exam Practice: Limits, Continuity, and Derivatives

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