Q: Which of the following statements about limits is TRUE?

If $$f(x) is $$continuous at $$x = a$$, then $$\lim_{x \to a} f(x) = f(a).$$

Question 1

Given the graph of $$f(x) = x^2$ (red curve in the first image), which line represents the tangent to the curve at x = 1$?

Accepted Answer

The green line

Question 2

What is the economic interpretation of the slope in the cost function $$C(x) = 2x + 1000$$?

Accepted Answer

It represents the marginal cost, or the cost to produce one additional unit.

Question 3

For the oil tank scenario, the equation $$y = 30,000 - 400t$$ models the gallons of oil remaining after $$t$$ days. What does the slope $$-400$$ represent?

Accepted Answer

The rate at which oil is being used per day.

Question 4

Which of the following is the correct point-slope form for a line passing through $$(2, -1)$$ with slope $$3$$?

Accepted Answer

$$y + 1 = 3(x - 2)$$

Question 5

Find the slope of the line passing through the points $$(6, -2)$$ and $$(9, -4).$$

Accepted Answer

$$-\frac{2}{3}$$

Question 6

Given $$f(x) = x^3$, what is f$$'(x)$$ using the power rule?

Accepted Answer

$$3x^2$$

Question 7

Evaluate the derivative of $$f(x) = \sqrt{x} at$$ $$x = 1.$$

Accepted Answer

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

Question 8

Which of the following statements about limits is TRUE?

Accepted Answer

If $$f(x) is $$continuous at $$x = a$$, then $$\lim_{x 	o a} f(x) = f(a).$$

Question 9

Using the difference quotient, what is the derivative of $$f(x) = x^2$?

Accepted Answer

$$2x$$

Question 10

Given the cost function $$C(x) = 6x^2$ + 2x + 10$$, what is the marginal cost when $$x = 10$$?

Accepted Answer

$$C'(10) = 122$$

Question 11

If $$s(t) = -16t^2$ + 128t + 5$$ represents the height of a ball in feet after $$t$$ seconds, what is the velocity at $$t = 2$$ seconds?

Accepted Answer

$$v(2) = 96$$

Question 12

Which of the following is NOT a reason a function might fail to be differentiable at a point?

Accepted Answer

The function is linear at that point.

Question 13

Suppose $$f(x) = x^2$ + 2x + 2. $$Use the difference quotient to find $$f'(x).$$

Accepted Answer

$$2x + 2$$

Question 14

If $$f(x) = x^4$ - 5x^3$ + 7$$, what is $$f''(x)$$?

Accepted Answer

$$12x^2 - 30x$

Question 15

Given the limit $$\lim_{x 	o \infty} \frac{1}{x^2$ + 1}$$, what is the value?

Accepted Answer

$$0$$

Question 16

If $$p(5000) = 300$$ and $$p'(5000) = 2$$, what does $$p'(5000) = 2$$ mean in the context of production?

Accepted Answer

Producing one more unit of labor increases output by approximately 2 units.

Question 17

Which of the following is the correct application of the constant multiple rule for differentiation?

Accepted Answer

$$\frac{d}{dx}[5x^3$] = 15x^2$

Question 18

If $$f(x) = x^3$ + \frac{1}{x}$$, what is $$f''(x)$$?

Accepted Answer

$$6x - 2x$$^{-3}$$

Question 19

A railroad company charges $$10 per mile up to 200 miles and 8 $$per mile after that, plus a $1000 base charge. What type of function best models the cost?

Accepted Answer

A piecewise linear function

Business Calculus: The Derivative – Foundations, Limits, and Applications

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