Q: Which of the following limits represents the definition of the derivative of $$f(x) at$$ $$x = a$$?

$$\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

Which of the following best describes the concept of a derivative as introduced in Calculus I?

Accepted Answer

The derivative of a function at a point is the slope of the tangent line to the function at that point.

Question 2

Suppose the position $$s(t) of a $$car in meters is given by $$s(t) = 4t^2$ + 2t + 1$$, where $$t is in $$seconds. What is the instantaneous velocity of the car at $$t = 3$$ seconds?

Accepted Answer

$$26 m/s$$

Question 3

Which of the following limits represents the definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\displaystyle \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 4

A pharmaceutical company models the rate at which a drug leaves the bloodstream by $$R(t) = 5e$$^{-0.2t}$$ mg/hour, where t$ is in hours. What is the total amount of drug eliminated from the bloodstream in the first 3$ hours?

Accepted Answer

$$\displaystyle \$$int_0$$^3 5e$$^{-0.2t}$$ \, dt$$

Question 5

Which of the following statements about continuity is TRUE?

Accepted Answer

A function $$f(x) is $$continuous at $$x = a if$$ $$\displaystyle \lim_{x 	o a} f(x) = f(a).$$

Question 6

Evaluate the following limit using L'Hôpital's Rule: $$\displaystyle \lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

1

Question 7

A company wants to minimize the cost $$C(x) = x^2$ - 8x + 20$$ for producing $$x$$ units. What is the minimum cost and at how many units does it occur?

Accepted Answer

Minimum cost is $$4 at$$ $$x = 4$$ units.

Question 8

Which of the following is the correct application of the Chain Rule for differentiating $$y = \sin(3x^2$)$$?

Accepted Answer

$$y' = \cos(3x^2$) \cdot 6x$$

Question 9

Suppose $$f(x) is $$differentiable and $$f'(x) \u003e 0$$ for all $$x in an $$interval. Which of the following must be true about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$increasing on that interval.

Question 10

A ball is thrown upward with an initial velocity of $$20 m/s $$from a height of $$50 m. $$Its height after $$t$$ seconds is $$h(t) = -4.9t^2$ + 20t + 50. At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = 2.04$$ seconds

Question 11

Which of the following is the correct anti-derivative of $$f(x) = 6x^2$?

Accepted Answer

$$2x^3 + C$

Question 12

A tank is being filled with water at a rate given by $$r(t) = 10 + 2t$$ liters per minute. How much water enters the tank in the first $$5$$ minutes?

Accepted Answer

$$\displaystyle \$$int_0$$^5 (10 + 2t) \, dt = 75$$ liters

Question 13

Which of the following is the correct statement of the Fundamental Theorem of Calculus?

Accepted Answer

If $$F(x) is an $$antiderivative of $$f(x) on$$ $$[a, b]$$, then $$\displaystyle \int_a^b f(x) \, dx = F(b) - F(a).$$

Question 14

Which of the following is the correct derivative of $$y = \ln(x^2$ + 1)$$?

Accepted Answer

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

Question 15

A company’s revenue function is $$R(x) = 100x - x^2$, where x$ is the number of units sold. What is the maximum revenue, and how many units should be sold to achieve it?

Accepted Answer

Maximum revenue is $$2500 at$$ $$x = 50$$ units.

Question 16

Which of the following is the correct result for $$\displaystyle \$$int_0$$^1 3x^2$ \, dx$$?

Accepted Answer

1

Question 17

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

Accepted Answer

$$2e^{2x}$$

Question 18

Which of the following is the correct application of implicit differentiation for the equation $$x^2$$ + $$y^2 = 25$?

Accepted Answer

$$2x + 2y \frac{dy}{dx} = 0$$

Question 19

A hospital monitors the temperature of a patient using the function $$T(t) = 37 + 2e$$^{-0.5t}$$, where t$ is in hours. What is the rate of change of temperature at t = 2$ hours?

Accepted Answer

$$-e^{-1}$$ degrees/hour

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