Q: Is the function $$h(x)$$ defined as $$h(x) = \begin{cases} \frac{x^2$ - 25}{x - 5} & x \neq 5 \\ 10 & x = 5 \end{cases}$$ continuous at $$x = 5$$? Justify your answer.

Yes, because $$\lim_{x \to 5} h(x) = 10$$ and $$h(5) = 10.$$

Question 1

Evaluate the following limit: $$\lim_{x 	o 2} \frac{5x^2$ - 20}{x - 2}$$

Accepted Answer

$$10$$

Question 2

Is the function $$h(x)$$ defined as $$h(x) = \begin{cases} \frac{x^2$ - 25}{x - 5} & x 
eq 5 \ 10 & x = 5 \end{cases}$$ continuous at $$x = 5$$? Justify your answer.

Accepted Answer

Yes, because $$\lim_{x 	o 5} h(x) = 10$$ and $$h(5) = 10.$$

Question 3

Find the derivative of $$f(x) = -x^2$ - 3x$$ using the formula for the derivative of a function $$y = f(x) at$$ $$x.$$

Accepted Answer

$$f'(x) = -2x - 3$$

Question 4

Differentiate $$y = 5x$$^{3/2}$$ 2x$$^{5/2}$$ - 12$$ with respect to $$x.$$

Accepted Answer

$$y' = 5 \cdot \frac{3}{2} x$$^{1/2}$$ \cdot 2x$$^{5/2}$$ + 5x$$^{3/2}$$ \cdot 2 \cdot \frac{5}{2} x$$^{3/2}$$ - 0$$

Question 5

Differentiate $$y = \frac{x^5$}{e^x}$$ with respect to $$x.$$

Accepted Answer

$$y' = \frac{5x^4$ e^x - x^5$ e^x}{e$$^{2x}$$}$$

Question 6

Find the derivative of $$y = (2x^7$ - 5x^2$)(-3x^2 + x$$)$.

Accepted Answer

$$y' = (2x^7$ - 5x^2$)'(-3x^2$ + x) + (2x^7$ - 5x^2$)(-3x^2 + x$$)'$

Question 7

Differentiate $$y = (9x^2$ - \ln(x$$))^{-100}$$ with respect to x$.

Accepted Answer

$$y' = -100(9x^2$ - \ln(x$$))^{-101}$$(18x - \frac{1}{x})$$

Question 8

Find the equation of the tangent line to the function $$f(x) = x^4$ at the point (2$$, 16)$$.

Accepted Answer

$$y = 32x - 48$$

Question 9

A company finds that the cost and revenue, in dollars, of producing $$x$$ hundred items are given by $$C(x) = 4x^2$ and R(x$$) = $$6x^2 + 2x - 1$, respectively. Determine the rate at which the company's profit is changing when 900 items have been produced and sold.

Accepted Answer

$$P'(9) = 14$$

Question 10

Differentiate $$f(x) = \frac{e$$^{3x}$$ + \ln(4x^2$)}{x^2$}$$ with respect to $$x.$$

Accepted Answer

$$f'(x) = \frac{3e$$^{3x}$$x^2$$ + \frac{2}{x}$$x^2$$ - $$2(e^{3x}$$ + \$$ln(4x^2$$))}{$$x^4$$}$

Question 11

Which of the following best describes the process for finding the derivative of a product of two functions $$u(x)$$ and $$v(x)$$?

Accepted Answer

$$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$

Question 12

If $$f(x) = x^3$ - 2x^2$ + 5x - 7$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 3x^2$ - 4x + 5$$

Question 13

A pharmaceutical company models the cost of producing $$x$$ batches of a drug as $$C(x) = 5x^2$ + 3x + 100. $$What is the marginal cost when $$x = 10$$?

Accepted Answer

$$C'(10) = 103$$

Question 14

If $$y = e$$^{2x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2e^{2x}$$

Question 15

A company’s revenue function is $$R(x) = 8x^2$ + 5x. $$What is the rate of change of revenue when $$x = 4$$?

Accepted Answer

$$R'(4) = 69$$

Question 16

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 17

Which of the following is the correct derivative of $$y = x^4$?

Accepted Answer

$$y' = 4x^3$

Question 18

A company’s profit function is $$P(x) = R(x) - C(x)$$, where $$R(x) = 10x^2$ + 3x$$ and $$C(x) = 7x^2$ + 2x. $$What is the marginal profit when $$x = 5$$?

Accepted Answer

$$P'(5) = 23$$

Business Calculus Study Guidance: Limits, Continuity, Derivatives, Tangents, and Applications

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