5 Steps to a 5 AP Calculus BC 2019

424 STEP 5. Build Your Test-Taking Confidence

29. The correct answer is (B).

1

n

[(

1

n

) 2

+

(

2

n

) 2

+···

(

n− 1

n

) 2 ]

represents the sum of the areas ofnrectangles

using LRAM each of width

1

n

. The heights of
the rectangles are the squares of the division
points, 0,

1

n

,

2

n

,

3

n

,...,

n− 1

n

, all of which are

between 0 and 1. Note that the first point from

the left isx=0. Thus,

nlim→∞

1

n

[(

1

n

) 2

+

(

2

n

) 2

+···

(

n− 1

n

) 2 ]

represents the area under they=x^2 from

0to1,or

∫ 1

0

x^2 dx.

30. The correct answer is (C).

V=π

∫ 1

0

(x−x^2 )^2 dx

=π

∫ 1

0

(x^4 − 2 x^3 +x^2 )dx

=π

[

x^5

5

−

x^4

2

+

x^3

3

] 1

0

=

π

30

Section I Part B

76. The correct answer is (C).

To assure thatf(x)=

{

ln(3−x)ifx< 2

a−bx ifx≥ 2

is differentiable atx=2, we must first be

certain that the function is continuous. As

x→2,

ln(3−x)→0, so we wanta− 2 b= 0

⇒a= 2 b. Continuity does not guarantee

differentiability, however; we must assure that

limh→ 0

f(2+h)−f(2)

h

exists. We must be certain

that limh→ 0 −

ln(3−(2+h))−ln(3−2)

h

is equal to limh→ 0 +

(a−b(x+h))−(a−bx)

h

.

hlim→ 0 −

ln(3−(2+h))−ln(3−2)

h

=hlim→ 0 −

ln(1−h)

h

=

0

0

. Thus, limh→ 0 −

(

1

1 −h

)

(−1)

=−1. limh→ 0 +

(a−b(2+h))−(a− 2 b)

h

=

−b⇒−b=− 1 ⇒b= 1 ⇒a=2.

77. The correct answer is (B).
    Sinceh(x)=f(g(x)),h′(x)= f′(g(x))g′(x)
    andh′(2)=f′(g(2))g′(2)=f′(3)(1)=−1.