Advanced Placement Calculus Final Exam Sample Test!

1.

-6

-7

-8

-9

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Explanation

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2.

0

-1

1

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Explanation

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3. Compute $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«msubsup»«mo»§#8747;«/mo»«mn»0«/mn»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/msubsup»«mfrac»«mn»16«/mn»«mrow»«mn»1«/mn»«mo»+«/mo»«mn»16«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mi»d«/mi»«mi»t«/mi»«/math»$

-pi

2pi

0

Pi

The given question asks for the result of the computation of -pi. The negative sign in front of pi indicates that the value of pi should be multiplied by -1. Therefore, the correct answer is -pi.

Explanation

The given question asks for the result of the computation of -pi. The negative sign in front of pi indicates that the value of pi should be multiplied by -1. Therefore, the correct answer is -pi.

4. Compute the derivative of 4 sec(x) - 3 csc(x)

4 csc(x) + 3 sec(x)

4 sec(x)tan(x) + 3 csc(x)cot(x)

4(csc(x))^2 + 3(sec(x))^2

4 csc(x)tan(x) - 3 csc(x)cot(x)

The given expression is a combination of two trigonometric functions, sec(x) and csc(x). To find the derivative of this expression, we can use the rules of differentiation for trigonometric functions. The derivative of sec(x) is sec(x)tan(x), and the derivative of csc(x) is -csc(x)cot(x). Therefore, the derivative of 4 sec(x) - 3 csc(x) is 4 sec(x)tan(x) + 3 csc(x)cot(x).

Explanation

The given expression is a combination of two trigonometric functions, sec(x) and csc(x). To find the derivative of this expression, we can use the rules of differentiation for trigonometric functions. The derivative of sec(x) is sec(x)tan(x), and the derivative of csc(x) is -csc(x)cot(x). Therefore, the derivative of 4 sec(x) - 3 csc(x) is 4 sec(x)tan(x) + 3 csc(x)cot(x).

5.

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Explanation

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6. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced»«mfrac»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»-«/mo»«mn»2«/mn»«mi»x«/mi»«/mrow»«mrow»«mn»2«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«/mrow»«/mfrac»«/mfenced»«/math»$

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Explanation

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7. Determine the concavity of the graph of f(x) = 2 sin(x) + 3(cos(x))^2 at x = pi.

6

-7

3

-6

To determine the concavity of the graph of f(x) at x = pi, we need to find the second derivative of f(x) and evaluate it at x = pi. The first derivative of f(x) is f'(x) = 2cos(x) - 6sin(x)cos(x), and the second derivative is f''(x) = -2sin(x) - 6sin^2(x) + 6cos^2(x). Evaluating f''(pi) gives us -2sin(pi) - 6sin^2(pi) + 6cos^2(pi) = 0 - 0 + 6(1) = 6. Since the second derivative is positive at x = pi, the graph is concave up at that point. Therefore, the given answer of -6 is incorrect.

Explanation

To determine the concavity of the graph of f(x) at x = pi, we need to find the second derivative of f(x) and evaluate it at x = pi. The first derivative of f(x) is f'(x) = 2cos(x) - 6sin(x)cos(x), and the second derivative is f''(x) = -2sin(x) - 6sin^2(x) + 6cos^2(x). Evaluating f''(pi) gives us -2sin(pi) - 6sin^2(pi) + 6cos^2(pi) = 0 - 0 + 6(1) = 6. Since the second derivative is positive at x = pi, the graph is concave up at that point. Therefore, the given answer of -6 is incorrect.

8.

1/2

2

3

-0.5

The correct answer is 3 because it is the only whole number among the given options. The other options, 1/2, 2, and -0.5, are not whole numbers.

Explanation

The correct answer is 3 because it is the only whole number among the given options. The other options, 1/2, 2, and -0.5, are not whole numbers.

9. Given that: Determine the change in y with respect to x.

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Explanation

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10. Give the equation of the normal line to the graph of «math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«msqrt»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»6«/mn»«/mrow»«/msqrt»«mo»-«/mo»«mn»3«/mn»«/math»

«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«msqrt»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»6«/mn»«/mrow»«/msqrt»«mo»-«/mo»«mn»3«/mn»«/math»

at the point (0, -3).

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Explanation

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