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 in Calculus?

Accepted Answer

The instantaneous rate of change of a function at a point.

Question 2

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

Accepted Answer

$$v(4) = 6(4) + 2 = 26 m/s$$

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 medication is administered to a patient and its concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the rate at which the concentration is changing at t = 2$ hours?

Accepted Answer

$$C'(2) = -5e$$^{-1}$$

Question 5

Which of the following statements about the definite integral $$\int_a^b f(x)\,dx is $$TRUE?

Accepted Answer

It represents the net area between the graph of $$f(x)$$ and the $$x-$$axis from $$x = a to$$ $$x = b.$$

Question 6

Evaluate the following limit: $$\displaystyle \lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

1

Question 7

A company’s profit $$P (in $$thousands of dollars) is given by $$P(x) = -2x^2$ + 12x - 16$$, where $$x is $$the number of units sold (in thousands). For what value of $$x is $$the profit maximized?

Accepted Answer

$$x = 3$$

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

If $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, find all critical points of $$f(x).$$

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 10

A tank is being filled with water at a rate given by $$R(t) = 5t^2$ liters per minute, where t$ is in minutes. How much water is added to the tank between t = 0$ and t = 3$ minutes?

Accepted Answer

$$\$$int_0$$^3 5t^2$\,dt = 45$$ liters

Question 11

Which of the following is the correct antiderivative of $$f(x) = 4x^3$?

Accepted Answer

$$x^4 + C$

Question 12

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

Accepted Answer

$$t = 2.04 s$$

Question 13

Which of the following statements about the Mean Value Theorem is correct?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, then there exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Question 14

Evaluate $$\$$int_1$$^2 \frac{1}{x} dx.$$

Accepted Answer

$$\ln 2$$

Question 15

A bacteria culture grows at a rate proportional to its size. If the initial population is $$1000$$ and after $$3$$ hours it is $$2000$$, what is the population after $$6$$ hours?

Accepted Answer

$$4000$$

Question 16

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

Accepted Answer

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

Question 17

A chemical reaction’s temperature (in $$^ C) is $$modeled by $$T(t) = 100 - 80e$$^{-0.2t}$$, where t$ is in minutes. What is the initial temperature of the reaction?

Accepted Answer

$$T(0) = 20^ 0C$$

Question 18

Which of the following is the correct application of the Product Rule for differentiating $$y = x^2$ e^x$$?

Accepted Answer

$$y' = 2x e^x + x^2$ e^x$$

Question 19

A patient’s heart rate is modeled by $$H(t) = 70 + 10\sin(\pi t/6)$$, where $$t is in $$minutes. What is the maximum heart rate predicted by this model?

Accepted Answer

$$80$$ bpm

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