Q: Which of the following is the correct limit definition of the derivative of $$f(x) at$$ $$x = a$$?

$$\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Q: A biologist models the population of bacteria by $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of change of the population at t = 10$ hours?

$$100 \times 0.05 \times e$$^{0.05 \times 10}$$

Q: Which of the following is the correct application of L'Hospital's Rule?

If $$\lim_{x \to a} \frac{f(x)}{g(x)} is$$ $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}$$, then $$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}.$$

Question 1

Which of the following statements is TRUE regarding the continuity of a function $$f(x) at$$ $$x = c$$?

Accepted Answer

The function $$f(x) is $$continuous at $$x = c if$$ $$f(c)$$ exists, $$\lim_{x 	o c} f(x)$$ exists, and $$\lim_{x 	o c} f(x) = f(c).$$

Question 2

A hospital monitors the temperature of a patient over time, modeled by a continuous function $$T(t). $$According to the Extreme Value Theorem, what can be concluded about $$T(t) on $$the interval $$[0, 24]$$ hours?

Accepted Answer

$$T(t)$$ must have an absolute maximum and minimum on $$[0, 24].$$

Question 3

Given $$f(x) = |x|$$, which statement about its differentiability is correct?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 0.$$

Question 4

A company’s profit $$P(x)$$ depends on the number of units $$x$$ sold. If $$P'(x) = 0$$ and $$P''(x) \u003c 0 at$$ $$x = 100$$, what does this indicate?

Accepted Answer

There is a local maximum at $$x = 100.$$

Question 5

Which of the following is the correct limit definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$$

Question 6

Suppose $$g(x) is $$differentiable and $$g(2) = 5$$, $$g'(2) = 3. $$What is the value of $$\left( g$$^{-1}$$ \right)'(5)$$?

Accepted Answer

$$\frac{1}{3}$$

Question 7

A biologist models the population of bacteria by $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of change of the population at t = 10$ hours?

Accepted Answer

$$100 	imes 0.05 	imes e$$^{0.05 	imes 10}$$

Question 8

Which of the following is an application of the Mean Value Theorem (MVT) in a real-world scenario?

Accepted Answer

If a car travels 100 miles in 2 hours, there must be a moment when its instantaneous speed is exactly 50 mph.

Question 9

Given $$f(x) = x^2$ + 3x$$, use the limit definition of the derivative to find $$f'(1).$$

Accepted Answer

$$5$$

Question 10

A medication’s concentration in the bloodstream is modeled by $$C(t) = 20 - 2t^2$. At what time t$ does the concentration reach its maximum?

Accepted Answer

$$t = 0$$

Question 11

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

Accepted Answer

Mean Value Theorem

Question 12

Which of the following functions is NOT differentiable at $$x = 0$$?

Accepted Answer

$$f(x) = |x|$$

Question 13

A chemical reaction rate is given by $$R(t) = t^3$ - 6t^2$ + 9t. $$Find the time $$t$$ when the reaction rate is at a local minimum.

Accepted Answer

$$t = 3$$

Question 14

Which rule should be used to differentiate $$y = (3x^2$ + 2x) \cdot e^x$$?

Accepted Answer

Product Rule

Question 15

If $$f(x) = \sin(x^2$)$$, which differentiation rule is required to find $$f'(x)$$?

Accepted Answer

Chain Rule

Question 16

A tank is being filled with water such that the volume $$V (in $$liters) at time $$t (in $$minutes) is $$V(t) = 5t^2$. What is the rate at which water is being added at t = 4$ minutes?

Accepted Answer

$$40$$ liters/min

Question 17

Which of the following is the correct application of L'Hospital's Rule?

Accepted Answer

If $$\lim_{x 	o a} \frac{f(x)}{g(x)} is$$ $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}$$, then $$\lim_{x 	o a} \frac{f(x)}{g(x)} = \lim_{x 	o a} \frac{f'(x)}{g'(x)}.$$

Question 18

A ball is thrown upward and its height at time $$t is$$ $$h(t) = -4.9t^2$ + 20t + 2. $$What is the maximum height reached by the ball?

Accepted Answer

$$22.4$$ meters

Question 19

Which statement about the second derivative $$f''(x) is $$correct?

Accepted Answer

If $$f''(x) \u003e 0$$, the function is concave up at $$x.$$

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