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# 12年级-AP微积分

International Department of Yaohua High School

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Grade 12nd Mid-Term Examination The 1st Semester of 2014-2015 Academic Year International Department of Yaohua High School
SUBJECT AP Calculus BC TIME 60 minutes
170t 2t ? 14

3. Find the limit (if it exists).

TOTAL SCORE 100 % where v is measured in feet per a)

?x ? 0

lim

? x ? ?x ?

2

+ 3 ? x ? ?x ? + 4 ? ? x 2 + 3x + 4 ? ?x

1. Suppose that an automobile's velocity starting from rest is v ? t ? ?

1 3 3 2 x + x + 4x 3 2

second. Find the acceleration at 5 seconds. Round your answer to one decimal place. a) b) c) d) e) 20.7 ft/sec2 1.5 ft/sec2 2.3 ft/sec2 4.1 ft/sec
2

b) c) d) e)

x3 + 3x 2 + 4 x

2x + 3
does not exist

x 2 + 3x + 4

0.5 ft/sec2

4. The volume V in liters of air in the lungs during a five-second respiratory cycle is approximated by the model V ? 0.1722t ? 0.1523t 2 ? 0.0375t 3 , where t is the time in seconds. Approximate the average volume of air in the lungs during one cycle. Round your answer to four decimal places.

2 2 2. Find d y / dx in terms of x and y given that 7 ? 4 xy ? 7 x ? 4 y.

--------------------------------

a)

d y ? 7 y2 dx 2
d2 y ? ?7 y3 2 dx

2

a) b) c) d)

0.8203 liter 0.5278 liter 2.0028 liter 1.8711 liter 0.4064 liter

b)

c)

d y ? ?4 y 2 2 dx

2

e)

NAME

d)

d2y ?0 dx 2
d2 y ? ?4 y3 dx2

e)

Mid-term Examination Grade 12 AP Calculus

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Mid-term Examination Grade 12 AP Calculus

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International Department of Yaohua High School

5. Find the cubic function of the form
f ? x ? ? ax3 ? bx2 ? cx ? d , where a ? 0 and the coefficients a, b, c, d are real numbers, which satisfies the

7. The graph of the function g ( y) ? y 3 is given below. Which of the following definite integrals yields the area of the shaded region?

conditions given below.

Relative maximum: ? 3,5 ? Relative minimum: ? 5,3 ? Inflection point: ? 4, 4 ?
a) b) c) d) e)
f ? x? ? f ? x? ? 1 3 41 x ? 6 x2 ? x ? 21 2 2 1 3 41 x ? 6 x2 ? x ? 21 2 2

1 45 f ? x ? ? ? x3 ? 6 x 2 ? x ? 22 2 2 f ? x? ? f ? x? ? 1 3 45 x ? 6 x 2 ? x ? 22 2 2 1 3 45 x ? 6 x 2 ? x ? 22 2 2

a)

6. Find the limit of s(n) as n ?? .

?

? y dy
3

0

1 ? n(n ? 1)(6n ? 1) ? s(n) ? 2 ? ? 7 n ? ?
a) b) c) d) e) unbounded 12/7 1/7 6/7 6

b)

? y dy
3 0

2

c)

?

? y dy
3

2

d)

? y dy
3 0

8

e)

? y dx
3 0

2

Mid-term Examination Grade 12 AP Calculus

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Mid-term Examination Grade 12 AP Calculus

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International Department of Yaohua High School

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8. Find the area of the shaded region for the function y ?

1 4 ? x2

.

10. Find the indefinite integral of the following function.

a)

? 2
2? 3

?

cos u du sin 6 u

b)

--------------------------------

c)

? 6
5? 6

d)

e)

? 3

NAME

9. Find an equation of the tangent line to the graph of y ? log 3 x at the point (9, 2) .

Mid-term Examination Grade 12 AP Calculus

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Mid-term Examination Grade 12 AP Calculus

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International Department of Yaohua High School

11. The rate of disbursement dQ / dt of a 2 million dollar federal grant is proportional to the square of
200 ? t. Time t is measured in days

12. An airplane flies at an altitude of 7 miles towards a point directly over an observer. Consider ? and x as shown in the following figure. The speed of the plane is 300 miles per hour. Find Round your answer to three decimal places.
d? when x ? 3 . dt

? 0 ? t ? 200? , and Q is the amount that remains to be disbursed.

Find the amount that remains to be disbursed after 50 days. Assume that all the money will be disbursed in 200 days.

Mid-term Examination Grade 12 AP Calculus

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Mid-term Examination Grade 12 AP Calculus

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