Q: A ball is dropped from a height of 240 meters and each bounce brings it to half its previous height. What is the total distance traveled by the ball before it stops?

$$240 + 2 \times 120 + 2 \times 60 + \ldots = 240 \left(1 + \frac{1}{1 - \frac{1}{2}}\right) = 720$$ meters

Question 1

Given the shaded region in the first image, which is bounded by $$y = x^4$ - 32$$ and $$y = -x^4$, what is the correct setup for the area of the region?

Accepted Answer

$$\int_{-2}^{2} \left[(x^4$ - 32) - (-x^4$)\right] dx$$

Question 2

In the second image, the shaded region is bounded by $$y = -x^4$ and y$$ = $$x^2 - 2x$. What is the area of the region between these curves from x = 0$ to x = 2$?

Accepted Answer

$$\int_{0}^{2} \left[(x^2$ - 2x) - (-x^4$)\right] dx$$

Question 3

Refer to the third image. What is the value of $$\lim_{x 	o \frac{\pi}{2}^+} f(x)$$ for the function shown?

Accepted Answer

3

Question 4

In the fourth image, the shaded region is bounded above by $$y = 4$$ and below by $$y = \sqrt{x}. $$What is the area of the region for $$x$$ from $$1 to$$ $$4$$?

Accepted Answer

$$\int_{1}^{4} (4 - \sqrt{x}) dx$$

Question 5

In the fifth image, the shaded region is bounded above by $$y = 5$$ and below by $$y = \sqrt{x}. $$What is the area of the region for $$x$$ from $$1 to$$ $$5$$?

Accepted Answer

$$\int_{1}^{5} (5 - \sqrt{x}) dx$$

Question 6

For the function $$y = e$$^{5x}$$, what is the first derivative with respect to x$?

Accepted Answer

$$5e^{5x}$$

Question 7

A ball is dropped from a height of 240 meters and each bounce brings it to half its previous height. What is the total distance traveled by the ball before it stops?

Accepted Answer

$$240 + 2 	imes 120 + 2 	imes 60 + \ldots = 240 \left(1 + \frac{1}{1 - \frac{1}{2}}ight) = 720$$ meters

Question 8

If the marginal cost function is $$MC(x) = 3x^2$ + 20x$$ and $$C(0) = 80$$, what is the cost function $$C(x)$$?

Accepted Answer

Question 9

For the function $$f(x, y) = 5x^4$ + 4x^3$y^4$$ + $$y^2$$ + $$5x^3 + 6y + 8$, what is the partial derivative f_x(x$$, y)$?

Accepted Answer

$$20x^3$$ + $$12x^2y^4$ + 15x^2$

Question 10

A store can sell 40 bicycles per week at $$800 each. For each 25 $$price reduction, 4 more bicycles are sold. If $$x is $$the number of $$25$$ reductions, what is the price function $$p(x)$$?

Accepted Answer

$$p(x) = 800 - 25x$$

Question 11

For the annuity problem: $$500 is $$deposited monthly for $$5$$ years at $$6\%$$ per annum compounded monthly. What is the common ratio $$r$$?

Accepted Answer

$$r = 1.005$$

Question 12

Given the demand function $$d(x) = 600 - 0.4x$$ and supply function $$s(x) = 0.2x$$, what is the break-even quantity?

Accepted Answer

$$x = 1000$$

Question 13

For the function $$y = 6x^5$ + 10x^3$ - 20x^2$ + 8$$, what is the second derivative $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

$$30x^4$$ + $$60x^2 - 40$

Question 14

A balloon used in surgical procedures is cylindrical with length $$90.0 mm. $$Find the rate of change of surface area with respect to radius when $$r = 0.030 mm.$$

Accepted Answer

$$180.12\Delta mm$^2/mm$$

Question 15

Use Simpson's Rule with $$n = 4$$ steps to estimate $$\int_{0}^{1} \frac{3}{1 + x^2$} dx. $$Which is the closest answer?

Accepted Answer

$$\frac{8011}{1700}$$

Question 16

Find the limit: $$\lim_{x 	o \infty} \frac{9x^3$ - 5x^2$ + 3x}{-x^3$ - 2x + 7}$$

Accepted Answer

$$-\frac{9}{1}$$

Question 17

For the function $$y = x$$^{200}$$, what is $$\frac{$$d^{200}y$$}{$$dx^{200}$$}$$?

Accepted Answer

$$200!$$

Question 18

A car's position is given by $$s = 80t - 5t^2$. When does the car come to a stop?

Accepted Answer

$$t = 8$$ seconds

Question 19

For the function $$f(x) = 1.2x^2$ - 4.7x + 10.8 ($$for $$1 \leq x \leq 5$$), at which year is the number of repairs minimized?

Accepted Answer

Year 2

Calculus Study Guide: Area Between Curves and Limits

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