AP Calculus AB Questions and FRQ

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Multiple Choice (1-10)

1. 

Use the properties of sigma notation and the summation formulas to evaluate 

the summation from i equals 1 to 10 of the quantity i squared plus 3 times i minus 1
 . (5 points)

 

138

540

136

925

2. 

Use geometry to evaluate 

the integral from 0 to 8 of the quantity 5 minus the absolute value of the quantity x minus 3, dx
 . (5 points)

 

10.5

12.5

23

25

3. 

Write the definite integral for the summation: 

the limit as n goes to infinity of the summation from k equals 1 to n of the product of the square of the quantity 1 plus k over n squared and 1 over n
 . (5 points)

 

the integral from x equals 0 to 1 of x squared, dx

the integral from x equals 1 to 2 of the quantity x plus 1 squared, dx

the integral from x equals 1 to 2 of x squared, dx

the integral from x equals 2 to 1 of x squared, dx

4. 

Find 

the derivative with respect to x of the integral from 2 to x squared of the quantity the natural log of the quantity t squared plus 1, dt
 . (5 points)

 

Cannot be found

ln(x2 + 1)

(2x)ln(x4 + 1)

(2x)ln(x4 + 1) - ln(5)

5. 

Find an antiderivative of 

x squared divided by 3 plus C
 . (5 points)

 

the quotient of x cubed and 9 plus C

the quotient of 2 times x and 3 plus C

x3 + C

None of these

6. 

Evaluate 

the integral of the product of secant x and tangent x, dx
 . (5 points)

 

Cannot be found

tan(x) + C

0

sec(x) + C

7. 

Evaluate the integral 

the integral of the product of x and the quantity x squared plus 1 raised to the 4th power, dx
 . (5 points)

 

the product of 1 tenth times the 5th power of the quantity x squared plus 1 plus C

the product of 1 tenth times x squared times the 5th power of the quantity x squared plus 1 plus C

the product of 1 fifth times the 5th power of the quantity x cubed plus x plus C

the product of 1 fifth times the 5th power of the quantity x squared plus 1 plus C

8. 

Find the antiderivative of 

the product of 12 times x squared and e raised to the x cubed power
 . (5 points)

 

4 e raised to the x cubed power plus C

the product of 4 times x cubed and e raised to the x cubed power plus C

None of these

Cannot be found

9. 

Use your calculator to evaluate 

the definite integral from negative 1 to 1 of 1 divided by the quantity x squared plus 1, dx
 . Give 3 decimal places for your answer. (5 points) 

_______________________

 

10. 

Suppose 

the integral from 1 to 6 of g of x, dx equals 10
 and 
the integral from 4 to 6 of g of x, dx equals negative 2
 , find the value of 
the integral from 1 to 4 of 2 times g of x, dx
 . (5 points)

 

4

6

12

24

Multiple Choice (1-30)

1. 

Which of the following sums does not equal the others? (4 points)

 

the sum from i equals 1 to 3 of i squared

the sum from i equals 1 to 2 of i cubed

the sum from i equals 1 to 4 of the quantity i plus 1

the sum from i equals 4 to 5 of the quantity 2 times i minus 2

2. 

Estimate the area under the curve f(x) = x2 from x = 1 to x = 5 by using four inscribed (under the curve) rectangles. Answer to the nearest integer. (4 points)

 

__________________

3. 

List x1, x2, x3, x4 where xi is the left endpoint of the four equal intervals used to estimate the area under the curve of f(x) between x = 4 and x = 6. (4 points)

 

4, 4.5, 5, 5.5

4.5, 5, 5.5, 6

4.25, 4.75, 5.25, 5.75

4, 4.2, 5.4, 6

4. 

Write the summation to estimate the area under the curve y = 2x2 + 1 from x = 0 to x = 4 using 4 rectangles and left endpoints. (4 points)

 

the summation from i equals 0 to 4 of the quantity 2 times i squared plus 1

the summation from i equals 1 to 3 of the quantity 2 times i squared plus 1

the summation from i equals 0 to 3 of the quantity 2 times i squared plus 1

the summation from i equals 1 to 4 of the quantity 2 times i squared plus 1

5. 

If the area under the curve of f(x) = 25 - x2 from x = 0 to x = 5 is estimated using five approximating rectangles and left endpoints, will the estimate be an underestimate or overestimate? (4 points)

 

Underestimate

Overestimate

The area will be exact

The area cannot be estimated with just five rectangles

6. 

The Riemann sum, 

the limit as the maximum of delta x sub i goes to infinity of the summation from i equals 1 to n of f of the quantity x star sub i times delta x sub i
 , is equivalent to 
the limit as n goes to infinity of the summation from i equals 1 to n of f of the quantity a plus i times delta x, times delta x
 with 
delta x equals the quotient of the quantity b minus a and n
.

 Write the integral that produces the same value as 

the limit as n goes to infinity of the summation from i equals 1 to n of the product of the quantity 1 plus 3 times i over n and 3 over n
 . (4 points)

 

 

the integral from 1 to 3 of the quantity x plus 1, dx

the integral from 1 to 4 of x, dx

the integral from 1 to 4 of the quantity 3 times x plus 1, dx

the integral from 1 to 3 of x, dx

7. 

Write the Riemann sum to find the area under the graph of the function f(x) = x2 from x = 1 to x = 5. (4 points)

 

the summation from i equals 1 to n of the product of the quantity squared of 1 plus 5 times i over n and 4 over n

the limit as n goes to infinity of the summation from i equals 1 to n of the product of the quantity squared of 1 plus 4 times i over n and 4 over n

the summation from i equals 1 to n of the product of the quantity squared of 4 times i over n and 4 over n

the limit as n goes to infinity of the summation from i equals 1 to n of the product of i over n quantity squared and 4 over n

8. 

Use your calculator to evaluate 

the limit from x equals e to e squared of the natural logarithm of x, dx
 . Give your answer to the nearest integer. (4 points) 

_______________________

 

9. 

Use geometry to evaluate 

the integral from 0 to 10 of the function f of x, dx
 for 
f of x equals 5 for x less than or equal to 5 and equals the quantity 10 minus x for x greater than 5
 . (4 points)

 

12.5

25

37.5

Cannot be found

10. 

Use geometry to evaluate 

the integral from 0 to 2 of the square root of the quantity 4 minus x squared, dx
 . (4 points)

 

pi divided by 2

π

2π

4π

11. 

Given that the antiderivative of 

f of x equals 1 divided by the quantity x squared plus 1
 is F(x) = tan-1(x) + C, evaluate 
the integral from negative 1 to 1 of the 1 divided by the quantity x squared plus 1, dx
 . (4 points)

 

pi

pi over 2

pi over 4

0

12. 

Evaluate 

the integral from 0 to 4 of the absolute value of the quantity x minus 3, dx
 . (4 points)

 

9.5

10

4

5

13. 

Given 

G of x equals the integral from 4 to x of the square root of the quantity 1 plus t squared, dt
 , find G '(x). (4 points)

 

the square root of the quantity 1 plus t squared

the square root of the quantity 1 plus x squared plus C

the square root of the quantity 1 plus x squared

the square root of the quantity 1 plus x squared minus the square root of 17

14. 

Find 

the derivative with respect to x of the integral from 1 to x squared of the natural logarithm of t, dt
 . (4 points)

 

2xln(x2)

one divided by x squared

2 divided by x

ln(x2)

15. 

Determine the interval on which f(x) = 

the square root of the quantity of x plus 2
 is integrable. (4 points)

 

(-∞, 2)

[-2, ∞)

(-∞,-2) U (-2, ∞)

All reals

16. 

Evaluate the integral: 

the integral of the quantity x cubed over 4 plus 2 times x squared over 3 minus 1, dx
 (4 points)

 

x squared over 2 plus 4 times x over 3 minus 1 over x plus C

3 times x squared over 4 plus 4 times x over 3 minus 1 over x plus C

x4 + 2x3 - x + C

x to the 4th power over 16 plus 2 times x cubed over 9 minus x plus C

17. 

Evaluate 

the integral of the quotient of the quantity x cubed plus x and x, dx
 . (4 points)

 

x - 1 + C

x squared over 2 minus x cubed over 3 plus C

the quotient of x to the 4th power minus x cubed and 4 times x squared plus C

x squared over 2 minus x plus C

18. 

Evaluate the integral: 

the integral of the quotient of sine cubed x and the quantity 1 minus cosine squared x, dx
 . (4 points)

 

-cos(x) + C

cos(x) + C

one half times sine squared x plus C

None of these

19. 

If f(x) and g(x) are continuous on [a, b], which one of the following statements is false? (4 points)

 

the integral from a to a of f of x, dx equals 0

the integral from a to b of the sum of f of x and g of x, dx equals the integral from a to b of f of x, dx plus the integral from a to b of g of x dx

the integral from a to b of f of x, dx equals 1 minus the integral from b to a of f of x, dx

the integral from a to b of 5 minus f of x, dx equals the integral from a to b of 5, dx minus the integral from a to b of f of x, dx

20. 

Evaluate the integral 

the integral of the cube root of x squared, dx
 . (4 points)

 

1 fifth times x raised to the 4 fifths power plus C

5 sixths times x raised to the 6 fifths power plus C

5 halves times x raised to the 3 halves power plus C

2 sevenths times x raised to the 7 halves power plus C

21. 

Evaluate the integral 

the integral of the product of x to the 5th power and the 9th power of x to the 6th power minus 4, dx
 . (4 points)

 

the product of 1 over 50 and the 10th power of x to the 6th power minus 4 plus C

the product of 1 over 60 and the 10th power of x to the 6th power minus 4 plus C

the product of 1 over 10 and the 10th power of x to the 6th power minus 4 plus C

the product of x to the 6th power over 6 and and the 9th power of x to the 7th power over 7 minus 4 times x plus C

22

Evaluate the integral 

the integral of the product of the sine cubed of 2 times x and the cosine of 2 times x, dx
 . (4 points)

 

one fourth times the 4th power of sine of 2 times x plus C

one fourth times the 4th power of sine of 2 times x plus C

2sin4(2x) + C

None of these

23.

Which of the following integrals cannot be evaluated using a simple substitution? (4 points)

 

the integral of 1 divided by the quantity x squared plus 1, dx

the integral of 1 divided by the quantity x squared plus 1, dx

the integral of x divided by the quantity x squared plus 1, dx

the integral of x cubed divided by the quantity x to the 4th power plus 1, dx

24. 

Evaluate 

the integral of the quotient of 3 times x squared and the square root of 1 minus x cubed, dx
 . (4 points)

 

negative 1 times the natural logarithm of the square root of 1 minus x cubed, plus C

-ln|1 - x3| + C

sin-1(x3) + C

negative 2 times the natural logarithm of the square root of 1 minus x cubed, plus C

25. 

Evaluate the integral 

the integral of the product of the quantity x times the square root of x plus 3, dx
 . (4 points)

 

x squared over 3 times the quantity x plus 3 raised to the three-halves power, plus C

the product of 2 times x over 3 times and the quantity x plus 3 raised to the three-halves power, plus C

2 fifths times the quantities x minus 2 and the quantity x plus 3 raised to the three halves power, plus C

the quotient of the quantity x minus 3 and 2 times the square root of the quantity x plus 3, plus C

26. 

Which of the following definite integrals could be used to calculate the total area bounded by the graph of y = 1 - x2 and the x-axis? ( 4 points)

 

the integral from 0 to 1 of the quantity 1 minus x squared, dx plus the integral from 1 to 2 of the quantity 1 minus x square, dx

the integral from 0 to 1 of the quantity 1 minus x squared, dx minus the integral from 1 to 2 of the quantity 1 minus x square, dx

the integral from 0 to 2 of the quantity 1 minus x squared, dx

times the integral from 0 to 1 of the quantity 1 minus x squared, dx

27. 

Suppose 

the integral from 2 to 8 of g of x, dx equals 13
 , and 
the integral from 6 to 8 of g of x, dx equals negative 3
 , find the value of 
2 plus the integral from 2 to 6 of g of x, dx
 . (4 points)

 

16

18

8

32

28. 

Evaluate the integral 

the integral from negative 1 to 1 of 2 times the absolute value of x, dx
 . (4 points)

 

-1

1

0

2

29. 

Use your graphing calculator to evaluate to three decimal places the value of 

the integral from negative 1 to 1 of the product 2 and the square root of 1 minus x squared over 2, dx
 . (4 points)

 

3.771

3.636

1.571

1.111

30. 

the integral from negative 2 to 1 of 1 divided by x to the 4th power equals negative 3 over 8.
 (4 points)

 

True

False

1. 

FRQ (1-5)

Using n = 4 equal-width rectangles, approximate 

the integral from negative 2 to 2 of the quantity x squared plus 8, dx
 . Use the left end-point of each sub-interval to determine the height of each rectangle. 

_______________________

2. 

Water leaks from a tank at the rate of r(t) gallons per hour. The rate decreased as time passed, and values of the rate at two-hour time intervals are shown in the table below. The total amount of water that leaked out is evaluated by a Riemann sum. Find the upper estimate (left end-points of each rectangle) for the total amount of water that leaked out by using five rectangles.

Give your answer with one decimal place. 

t (hr)

0

2

4

6

8

10

r(t) (gal/hr)

10.7

8.6

6.6

5.2

5.0

4.5

_______________________

3. 

Find the interval on which the curve of 

y equals the integral from 0 to x of 6 divided by the quantity 1 plus 2 times t plus t squared, dt
 is concave up.

_______________________

4. 

Evaluate 

the integral of the quotient of the cosine of x and the square root of the quantity 1 plus sine x, dx
 .

_______________________

5. 

Evaluate exactly the value of 

the integral from negative 1 to 0 of the product of the cube of the quantity 2 times x to the 4th power plus 8 times x and 4 times x to the 3rd power plus 4, dx
 . Your work must include the use of substitution and the antiderivative. 

_______________________