Q: A biologist observes that a single-cell creature triples its population every 4 hours. If the initial population is 100, what is the population after 12 hours? (Assume exponential growth.)

$$100 \times 3$$^{3}$$ = 2700$$

Q: A ladder 10 ft long rests against a vertical wall. If the bottom slides away from the wall at 2 ft/sec, how fast is the top sliding down when the bottom is 6 ft from the wall?

$$-\frac{6}{8} \times 2 = -1.5 ft/sec$$

Question 1

A biologist observes that a single-cell creature triples its population every 4 hours. If the initial population is 100, what is the population after 12 hours? (Assume exponential growth.)

Accepted Answer

$$100 	imes 3$$^{3}$$ = 2700$$

Question 2

Evaluate the definite integral $$\int_{0}^{1} \sqrt{1 - x$$^{2}$$} \, dx$$ using geometric interpretation.

Accepted Answer

$$\frac{\pi}{4}$$

Question 3

Which of the following is the correct statement of the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$F(x) = \int_{a}^{x} f(t) \, dt$$, then $$F'(x) = f(x).$$

Question 4

A radioactive isotope decays according to $$\frac{dN}{dt} = -kN. If $$the half-life is 10 years, what is the value of $$k$$?

Accepted Answer

$$k = \frac{\ln 2}{10}$$

Question 5

Find the slope of the tangent to the curve $$y = x^2$ \sin x at $$the point $$(\pi, 0).$$

Accepted Answer

$$2\pi \sin \pi + \pi^2$ \cos \pi = 0 - \pi^2$ = -\pi^2$

Question 6

Which of the following is the correct formula for the linear approximation of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$L(x) = f(a) + f'(a)(x - a)$$

Question 7

Suppose $$F(x) = \int_{0}^{x} \sqrt{1 + t^2$} \, dt. $$What is $$F'(2)$$?

Accepted Answer

$$\sqrt{1 + 2^2$} = \sqrt{5}$$

Question 8

A ladder 10 ft long rests against a vertical wall. If the bottom slides away from the wall at 2 ft/sec, how fast is the top sliding down when the bottom is 6 ft from the wall?

Accepted Answer

$$-\frac{6}{8} 	imes 2 = -1.5 ft/sec$$

Question 9

Evaluate $$\lim_{x 	o 0} \frac{\sin(3x)}{x}.$$

Accepted Answer

$$3$$

Question 10

Which of the following functions is continuous everywhere?

Accepted Answer

$$f(x) = e$$^{x}$$

Question 11

Find the absolute maximum and minimum values of $$f(x) = x^3$ - 3x on $$the interval $$[-2, 2].$$

Accepted Answer

Question 12

Given $$f(x) = \frac{x^2$ - 1}{x - 1}$$, what is $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

$$2$$

Question 13

A Newton window is a window in the shape of a rectangle surmounted by a semicircle whose diameter equals the base of the rectangle. What is the largest possible area of a Newton window with perimeter $$8 m$$?

Accepted Answer

The area is maximized when the diameter of the semicircle equals the width of the rectangle, and can be found by maximizing $$A = w h + \frac{1}{2} \pi \left(\frac{w}{2}\$$right)^2$$ subject to 2h + w$$ + \pi \frac{w}{2} = 8$$.

Question 14

Find $$\frac{d}{dx} \left( \int_{0}^{x^2$} \sin t \, dt \right).$$

Accepted Answer

$$2x \sin(x^2$)$$

Question 15

Suppose $$f(x) is $$differentiable and $$f(2) = 3$$, $$f'(2) = -1. $$Find the linear approximation of $$f(x) at$$ $$x = 2.$$

Accepted Answer

$$L(x) = 3 - 1(x - 2)$$

Question 16

Which of the following is the correct application of the Substitution Rule for $$\int 2x \cos(x^2$) \, dx$$?

Accepted Answer

Let $$u = x^2$, then du = 2x$$ \, dx$$, so $$\int 2x \$$cos(x^2$$) \, dx = \int \cos(u) \, du$$

Question 17

Find $$\lim_{n 	o \infty} \sum_{i=1}^{n} \frac{1}{n} \sqrt{1 - \left( \frac{i}{n} \$$right)^2$$}.$$

Accepted Answer

$$\int_{0}^{1} \sqrt{1 - x^2$} \, dx$$

Question 18

Given $$f(x) = \sqrt{x}$$, what is the linear approximation at $$x = 4$$?

Accepted Answer

$$L(x) = 2 + \frac{1}{4}(x - 4)$$

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