Q: If $$\lim_{x \to a} f(x)$$ exists, which of the following must be true?

The left and right limits of $$f(x) as$$ $$x \to a$$ are equal

Q: An investor puts $$9900 into an account that pays 9% interest compounded continuously. How much will be in the account after 4 years? (Use A$$ = $$Pe^{rt}$)

$$A = 9900e$$^{0.09 \times 4}$$ \approx 14100$$

Q: If $$f is $$continuous at $$x = -10$$, which of the following must be true?

$$\lim_{x \to -10} f(x)$$ exists and equals $$f(-10)$$

Q: Which of the following statements is true about the function $$f(x) if$$ $$\lim_{x \to 2} f(x)$$ exists?

The left and right limits as $$x \to 2$$ are equal

Q: Suppose $$H(x) is $$defined everywhere except at $$x = -4. $$Which of the following is true?

$$\lim_{x \to -4} H(x)$$ may exist even if $$H(-4)$$ does not$

Question 1

Which of the following equations represents a function?

Accepted Answer

$$y = 3x^2$ - 9x - 66$$

Question 2

Given $$f(x) = x^2$ - 8x + 7$$, what is the vertex of the parabola?

Accepted Answer

$$(4, -9)$$

Question 3

Find the domain of the function $$f(x) = \frac{x-2}{6x-12}.$$

Accepted Answer

$$x 
eq 2$$

Question 4

If $$\lim_{x 	o a} f(x)$$ exists, which of the following must be true?

Accepted Answer

The left and right limits of $$f(x) as$$ $$x 	o a$$ are equal

Question 5

An investor puts $$9900 into an account that pays 9% interest compounded continuously. How much will be in the account after 4 years? (Use A$$ = $$Pe^{rt}$)

Accepted Answer

$$A = 9900e$$^{0.09 	imes 4}$$ \approx 14100$$

Question 6

Let $$f(x) = 2x^2$ - 9x + 79. $$What is $$f'(x)$$?

Accepted Answer

$$4x - 9$$

Question 7

Find the equation of the tangent line to $$f(x) = 3x^2$ - 9x - 66 at $$the point $$(6, -12).$$

Accepted Answer

$$y = 27x - 174$$

Question 8

If $$C(x) = 77x + 1900 is $$the cost and $$R(x) = 146x is $$the revenue for $$x$$ keyboards, what is the profit from selling $$x = 80$$ keyboards?

Accepted Answer

$$P(80) = 11680 - 8060 = 3620$$

Question 9

Given $$P(t) = 0.6t^2$ + 0.4t + 75$$, where $$t is $$the number of years since 2010, what is the average rate of change of $$P$$ between 2015 and 2024?

Accepted Answer

$$\frac{P(14) - P(5)}{14 - 5} = \frac{0.6(14^2$) + 0.4(14) + 75 - [0.6(5^2$) + 0.4(5) + 75]}{9} \approx 7.6$$

Question 10

Evaluate $$\lim_{x 	o \infty} \frac{5x^2$ - 9x + 8}{4x^2$ - 8x + 1}.$$

Accepted Answer

$$\frac{5}{4}$$

Question 11

If $$f is $$continuous at $$x = -10$$, which of the following must be true?

Accepted Answer

$$\lim_{x 	o -10} f(x)$$ exists and equals $$f(-10)$$

Question 12

Which of the following statements is true about the function $$f(x) if$$ $$\lim_{x 	o 2} f(x)$$ exists?

Accepted Answer

The left and right limits as $$x 	o 2$$ are equal

Question 13

Let $$f(x) = x^2$ - 8x + 7. $$What is the axis of symmetry?

Accepted Answer

$$x = 4$$

Question 14

For the function $$f(x) = x^2$ - 8x + 7$$, does the parabola open up or down?

Accepted Answer

Up

Question 15

Given $$f(x) = x^2$ - 8x + 7$$, what is the $$y-$$intercept?

Accepted Answer

$$7$$

Question 16

If $$f(-10) = -3$$ and $$f(2) = -3$$, is $$f a $$function?

Accepted Answer

Yes, because each input has only one output

Question 17

If $$\lim_{x 	o 8} f(x) = f(8)$$, what does this imply?

Accepted Answer

$$f is $$continuous at $$x = 8$$

Question 18

If $$f is $$continuous at $$x = -10$$, does $$f'(-10)$$ necessarily exist?

Accepted Answer

No, because continuity does not guarantee differentiability

Question 19

Suppose $$H(x) is $$defined everywhere except at $$x = -4. $$Which of the following is true?

Accepted Answer

$$\lim_{x 	o -4} H(x)$$ may exist even if $$H(-4)$$ does not$

Business Calculus: Functions, Limits, Continuity, and Applications Study Guide

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