Calc Multiple Choice

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1. 

Find the particular solution to y ′ = 3sin(x) given the general solution is y = C − 3cos(x) and the initial condition y(π) = 1. (5 points)

 

4 - 3cos(x)

-2 - 3cos(x)

2 - 3cos(x)

-4 - 3cos(x)

2. 

The slope of the tangent to a curve at any point (x, y) on the curve is negative 1 times x divided by y . Find the equation of the curve if the point (2, -2) is on the curve. (5 points)

 

x + y = 0

x2 - y2 = -2

x2 + y2 = 16

x2 + y2 = 8

3. 

The rate of decay in the mass, M, of a radioactive substance is given by the differential equation dM dt equals negative 1 times k times M , where k is a positive constant. If the initial mass was 150g, then find the expression for the mass, M, at any time t. (5 points)

 

M = e-kt

M = 150 ekt

M = 150 e-kt

M = 150ln(kt)

4. 

The temperature of a cup of hot tea varies according to Newton's Law of Cooling: dT dt equals negative k times the quantity T minus A , where T is the temperature of the tea, A is the room temperature, and k is a positive constant. If the water cools from 100°C to 80°C in 1 minute at a room temperature of 60°C, find the temperature, to the nearest degree Celsius of the coffee after 4 minutes. (5 points)

 

42

58

63

79

5. 

The differential equation dy dx equals the quotient of the quantity x minus 3 and x plus 4 (5 points)

I.produces a slope field with horizontal tangents at x = -4 II.produces a slope field with vertical tangents at x = -4 III.produces a slope field with columns of parallel segments

 

I only

II only

I and II

II and III only

6. 

Which of the following differential equations is consistent with the following slope field?

In quadrant one, all slopes are positive. Greater values of y have slopes approaching horizontal and y less than 1 have slopes approaching vertical. For quadrant two slopes are negative with slopes approaching horizontal for greater values of y. Quadrant three also has all negative slopes. Larger negative values of y have slopes approaching horizontal. All slopes are positive in quadrant four. (5 points)

 

dy over dx equals x divided by y squared

dy over dx equals x divided by y

dy over dx equals x squared divided by y

dy over dx equals x squared divided by y squared

7. 

The general solution of the differential equation x dx - y dy = 0 is a family of curves. These curves are all (5 points)

 

hyperbolas

circles

parabolas

ellipses

8. 

Estimate the value of the integral from negative 2 to 4 of x cubed, dx by using the Trapezoidal Rule with n = 3. (5 points)

 

252

128

63

72

9. 

The table below gives selected values for the function f(x). With 5 rectangles, using the left side of each rectangle to evaluate the height of each rectangle, estimate the value of the integral from 1 to 2 of f of x, dx . (5 points)

x

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

f(x)

1

0.909

0.833

0.769

0.714

0.667

0.625

0.588

0.556

0.526

0.500

 

0.7456

0.6456

0.6919

0.6932

10. 

Given f(x) > 0 with f ′ (x) > 0, and f ″(x) > 0 for all x in the interval [0, 3] with f(0) = 0.1 and f(3) = 1, the left, right, trapezoidal, and midpoint rule approximations were used to estimate the integral from 0 to 3 of f of x, dx. The estimates were 0.8067, 0.9635, 1.0514, 1.0753 and 1.3439, and the same number of subintervals were used in each case. Match the rule to its estimate. (5 points)

 

____

1.

left endpoint

____

2.

right endpoint

____

3.

midpoint

____

4.

trapezoidal

____

5.

actual area

a.

1.0753

b.

0.9635

c.

1.0514

d.

1.3439

e.

0.8067

11.

 

The graph of f ′(x) is continuous and decreasing with an x-intercept at x = 0. Which of the following statements is true? (4 points)

 

The graph of f has a relative maximum at x = 0.

The graph of f has a relative minimum at x = 0.

The graph of f has an inflection point at x = 0.

The graph of f has an x-intercept at x = 0.

12. 

The graph below shows the graph of f (x), its derivative f ′(x), and its second derivative f ″(x). Which of the following is the correct statement?

A graph is shown with three functions. Function A is a parabola with a minimum of about 0.25 comma 0.75 crossing the y axis just below 1. Function B is a cubic function crossing the origin increasing from left to right. Function C is a quartic function increasing to a local maximum at about negative 1.5 comma 0.75 then decreasing to an inflection point at about negative 0.25 comma 0.25 and then increasing to another local maximum at about 0.8 comma 0.25 and then decreasing down to the right. (4 points)

 

A is f ″, B is f ′, C is f.

A is f ″, B is f, C is f ′.

A is f ′, B is f, C is f ″.

A is f, B is f ′, C is f ″.

13. 

Below is the graph of f ′(x), the derivative of f(x), and has x-intercepts at x = -3, x = 1 and x = 2. There are horizontal tangents at x = -1.5 and x = 1.5. Which of the following statements is true?

Graph of a function that increases from negative infinity to x equals negative 1.5, decreases from x equals negative 1.5 to x equals 1.5, crossing the y axis at y equals 6, and increases from x equals 1.5 to positive infinity with x intercepts at x equals negative 3, 1 and 2. (4 points)

 

f has an inflection point in the interval x = -1 to x = 1.

f is increasing on the interval from x = -3 to x = 1.

f has a relative maximum at x = -1.5.

None of these is true.

14. 

The graph of f ′ (x), the derivative of f of x, is continuous for all x and consists of five line segments as shown below. Given f (0) = 7, find the absolute minimum value of f (x) over the interval [-3, 0].

Graph of line segments increasing from x equals negative 4 to x equals negative 3, decreasing from x equals negative 3 to x equals 0, increasing from x equals 0 to x equals 3, constant from x equals 2 to x equals 4 and decreases from x equals 4 to x equals 5. x intercepts at x equals negative 4, x equals 0, x equals 5. (4 points)

 

0

2.5

4.5

11.5

15. 

The graph of y = f ′(x), the derivative of f(x), is shown below. Given f(-4) = 2, evaluate f(4).

Graph consists of 3 line segments from x equals negative 4 to x equals 4. Graph is decreasing from x equals negative 4 to x equals negative 2, increases from x equals negative 2 to x equals 2 and decreases from x equals 2 to x equals 4. There are x intercepts at x equals negative 4, 0 and 4. (4 points)

 

0

2

8

10

16. 

Which of the following functions grows the fastest as x goes to infinity? (4 points)

 

2x

ln(x)

sin(x)

x20

17. 

Compare the rates of growth of f(x) = the square root of x and g(x) = Ln(x) as x approaches infinity. (4 points)

 

f(x) grows faster than g(x) as x goes to infinity.

g(x) grows faster than f(x) as x goes to infinity.

f(x) and g(x) grow at the same rate as x goes to infinity.

The rate of growth cannot be determined.

18. 

What does limit as x goes to infinity of the quotient of f of x and g of x equals 5 show? (4 points)

 

g(x) grows faster than f(x) as x goes to infinity.

f(x) and g(x) grow at the same rate as x goes to infinity.

f(x) grows faster than g(x) as x goes to infinity.

L'Hôpital's Rule must be used to determine the true limit value.

19. 

Which of the following functions grows at the same rate as the square root of the quantity x raised to the 4th power plus x ? (4 points)

 

x

x2

x3

x4

20. 

Which of the following functions grows the slowest as x goes to infinity? (4 points)

 

5x

5x

x5

They all grow at the same rate.

21. 

The function f is continuous on the interval [3, 13] with selected values of x and f(x) given in the table below. Use the data in the table to approximate f ′(12). (4 points)

x

3

4

7

11

13

f(x)

2

8

10

12

22

 

 

22. 

f is a differentiable function on the interval [0, 1] and g(x) = f(4x). The table below gives values of f '(x). What is the value of g '(0.1)? (4 points)

x

0.1

0.2

0.3

0.4

0.5

f '(x)

1

2

3

-4

5

 

-16

-4

4

Cannot be determined

23. 

f(x) and g(x) are a differentiable function for all reals and h(x) = g[f(3x)]. The table below gives selected values for f(x), g(x), f '(x), and g '(x). Find the value of h '(1). (4 points)

x

1

2

3

4

5

6

f(x)

0

3

2

1

2

0

g(x)

1

3

2

6

5

0

f '(x)

3

2

1

4

0

2

g '(x)

1

5

4

3

2

0

 

24. 

The table of values below shows the rate of water consumption in gallons per hour at selected time intervals from t = 0 to t = 12.

Using a right Riemann sum with 5 subintervals estimate the total amount of water consumed in that time interval. (4 points)

x

0

2

5

7

11

12

f(x)

5.7

5.0

2.0

1.2

0.6

0.4

 

2.742

21.2

32.9

None of these

25. 

The function is continuous on the interval [10, 20] with some of its values given in the table above. Estimate the average value of the function with a Right Hand Sum Approximation, using the intervals between those given points. (4 points)

x

10

12

15

19

20

f(x)

-2

-5

-9

-12

-16

 

-9.250

-10.100

-7.550

-6.700

26. 

Let F of x equals the integral from 0 to 2 times x of the tangent of t squared . Use your calculator to find F"(1). (4 points)

 

5.774

11.549

18.724

37.449

27. 

Cleaning pumps remove oil at the rate modeled by the function R, given by r of t equals 2 plus the cosine of the quantity pi times t divided by 11 with t measure in hours and and R(t) measured in gallons per hour. How much oil will the pumping stations remove during the 6-hour period from t = 0 to t = 6? Give 3 decimal places. (4 points) 

 

28. 

A particle moves along the x-axis with velocity v(t) = t2 - 1, with t measured in seconds and v(t) measured in feet per second. Find the total distance travelled by the particle from t = 0 to t = 2 seconds. (4 points)

 

0.667

2

4

None of these

29. 

Find the range of the function f of x equals the integral from negative 4 to x of the square root of the quantity 16 minus t squared . (4 points)

 

[-4, 4]

[-4, 0]

[0, 4π]

[0, 8π]

30. 

Use the graph of f(t) = 2t + 3 on the interval [-3, 6] to write the function F(x), where f of x equals the integral from 3 to x of f of t dt . (4 points)

 

F(x) = 2x2 + 6x

F(x) = 2x + 3

F(x) = x2 + 3x + 54

F(x) = x2 + 3x - 18

SHOW WORK (1-5)

1. 

Write and then solve for y = f(x) the differential equation for the statement: "The rate of change of y with respect to x is inversely proportional to y4."

_________________________________

2. 

Solve the differential equation dy dx equals the quotient of y squared and x cubed for y = f(x) with the condition y(1) = 1.

_________________________________

3. 

1. Solve the differential equation y prime equals the product of 4 times x and the square root of the quantity 1 minus y squared .

_________________________________

2. b. Explain why the initial value problem y prime equals the product of 4 times x and the square root of the quantity 1 minus y squared with y(0) = 4 does not have a solution.

_________________________________

4. 

The table below gives selected values for the function f(x). Use a trapezoidal estimation, with 6 trapezoids to approximate the value of the integral from 1 to 2 of f of x, dx. Give 3 decimal places for your answer. (10 points)

x

1

1.1

1.3

1.6

1.7

1.8

2.0

f(x)

1

3

5

8

10

11

14

_________________________________

5. 

Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for the integral from 0 to 8 of x squared, dx . (10 points)

_________________________________