$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Question 1

Which of the following correctly expresses the definition of the derivative of a function $$f(x) at a $$point $$x$$?

Accepted Answer

$$f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$

Question 2

Given $$f(x) = 15$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 0$$

Question 3

Using the power rule, what is the derivative of $$f(x) = 4x^3$ - 3x^2$ + 5x$$^{-3}$$?

Accepted Answer

$$f'(x) = 12x^2$ - 6x + 5x$$^{-4}$$

Question 4

If $$f(x) = 6 - 2x$$, what is the slope of the tangent line at $$x = 2$$?

Accepted Answer

$$2$$

Question 5

For the function $$f(x) = 6 - 2x$$, at what value of $$x is $$the tangent line horizontal?

Accepted Answer

$$x = 3$$

Question 6

Given $$f(x) = 6 - 2x$$, what is the equation of the tangent line at $$x = 2$$?

Accepted Answer

$$y = 2x + 4$$

Question 7

If $$S(t) = 0.03t^3$ + 0.5t^2$ + 2t + 3$$ represents a company's total sales (in millions of dollars) $$t$$ months from now, what is $$S'(t)$$?

Accepted Answer

$$S'(t) = 0.09t^2$ + t + 2$$

Question 8

Using $$S(t) = 0.03t^3$ + 0.5t^2$ + 2t + 3$$, what is the instantaneous rate of change of sales at $$t = 5$$ months?

Accepted Answer

$$S'(5) = 0.09(25) + 5 + 2 = 9.25$$

Question 9

If $$S(10) = 0.03(1000) + 0.5(100) + 2(10) + 3$$, what is the total sales in millions of dollars after 10 months?

Accepted Answer

$$S(10) = 30 + 50 + 20 + 3 = 103$$

Question 10

Which rule allows you to find the derivative of $$f(x) = a \cdot g(x)$$, where $$a is a $$constant?

Accepted Answer

The Constant Multiple Rule

Question 11

If $$f(x) = x^n$$, what is $$f'(x)$$ according to the power rule?

Accepted Answer

$$f'(x) = n x$$^{n-1}$$

Question 12

Suppose $$f(x) = x^4$ + 2x^2$ - 5x + 7. $$What is $$f'(x)$$?

Accepted Answer

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

Question 13

A company’s sales function is $$S(t) = 0.03t^3$ + 0.5t^2$ + 2t + 3. $$What does $$S'(t)$$ represent in a business context?

Accepted Answer

The instantaneous rate of change of sales at time $$t$$

Question 14

If $$f(x) = 6 - 2x$$, which of the following statements is true about the graph of $$f(x)$$?

Accepted Answer

It is a straight line with a negative slope

Question 15

Given $$f(x) = 4x^3$ - 3x^2$ + 5x$$^{-3}$$, what is the value of f$$'(1)$$?

Accepted Answer

$$f'(1) = 12 - 6 + 5 = 11$$

Question 16

Which property of derivatives allows you to differentiate $$f(x) + g(x) as$$ $$f'(x) + g'(x)$$?

Accepted Answer

The Sum Rule

Question 17

If $$f(x) = 2x^2$ + 3x$$, what is the equation of the tangent line at $$x = 1$$?

Accepted Answer

$$y = 7x - 2$$

Question 18

If $$f(x) = 6 - 2x$$, what is the $$y-$$intercept of the tangent line at $$x = 2$$?

Accepted Answer

$$4$$

Fundamentals of Derivatives and Tangent Lines in Business Calculus

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