BC Calc Test 5b
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All problems are to be solved algebraically and proper notation must be used.
1)_______
Multiple Choice- Write the letter of your choice in the space at right.
1) Suppose

0
3
f ( x)dx  6,

7
3
f ( x)dx  8, and

7
0
2)_______
g ( x)dx  10.
3)_______
4)_______
Which of the following are true?
I.
  f ( x)  g ( x)  dx  4
7
0
A) I only
B) II only
II.
  f ( x)  g ( x)  dx  10
7
0
C) III only
D) I and II
III.
  f ( x)  g ( x)  dx  12
7
0
E) I and III
2) The graph of a twice differentiable function f is shown in the figure at right.
Which of the following is true?
A) f (1)  f (1)  f (1)
B) f (1)  f (1)  f (1)
C) f (1)  f (1)  f (1)
.
1
D) f (1)  f (1)  f (1)
E) f (1)  f (1)  f (1)
3) If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is
decreasing at a rate of 3 inches per minute, which of the following must be true about the area A
of the triangle?
A) A is always increasing.
B) A is always decreasing. C) A is decreasing only when b < h.
D) A is decreasing only when b > h.
E) A remains constant.
4) The function h( x)  x 4  bx 2  8x  1 has a horizontal tangent and a point of inflection for the
same value of x. What must be the value of b?
A) –6
B) –1
C) 1
D) 4
E) 6
Complete this page without the use of your calculator.
5) Evaluate the following definite integrals.
a)

4
-1
e3 x dx
b)


0
cos 2 x  6 x 2 
5
dx
x 1
(  1) x  2  1 x  2
6) Let f be the function defined by f ( x)  
x2
sin  x  x  1
a) Show that f is differentiable at x = 2.
b) Find all 4 values of c that satisfy the mean value theorem over the interval [0, 4].
c) Set up, but do not evaluate, and integral expression that gives the average value of the
function over the interval [0, 4].
d) On some appropriate interval, I, the inverse of f exists. Let g ( x)  f 1 ( x) . Write the equation
of the line tangent to g at x = 5.
7) Let g ( x)  2 xe( x2)  x 2  8 as shown at right. Let f represent
the position of a particle on a number line. Define f as
t
follows: f (t )  5   g ( x)dx for time 0  t  8.
3
a) When is the particle moving to the left?
b) Write the equation for the line tangent to the graph of f at time t = 3.
c) Use your tangent line to approximate the position of the particle at time t = 3.1.
d) Without calculating the actual value of f (3.1), determine whether your approximation over or
underestimates the true value. Briefly explain your answer.
e) Where is the particle at time t = 7?
f) How far has the particle travelled in the first 7 seconds?
g) In the first 3 seconds, what is the maximum distance the particle is from the origin?
8) Let f be a function with the following properties:
i) f ( x)  ax 2  bx
ii) f (1)  6
iii) f (1)  18
iv)

2
1
f ( x)dx  18
Find an algebraic expression for f ( x ).
9) Let F ( x)  3 x 2  7 and F (1)  4.
a) Write an integral expression for F ( x).
b) Find F (8) .
c) Write an equation of the line tangent to F ( x) at x = 1.
d) Write an equation of line
such that line
intersects F ( x) when x = 1 and x = 8.
e) Write the equation of the tangent to F that is parallel to line
guarantees that such a line exists?
on the interval (1,8). What
Discarded, unwanted, unloved test questions.
1) Let f be a function defined on the closed interval [0, 7]. The
graph of f, consisting of four line segments, is shown at right.
x
Let g be the function given by g ( x)   f (t )dt.
2
a) Find g (3), g (3), and g (3).
b) Find the average rate of change of g on the interval 0  x  3.
c) For how many values c, where 0  c  3, is g (c) equal to the average rate found in part
b)? Explain your reasoning.
d) Find the x-coordinate of each point of inflection of the graph of g on the interval
0  x  7. Justify your answer.
2)
t
(minutes)
v(t)
(miles per minute
0
5
10
15
20
25
30
35
40
7.0
9.2
9.5
7.0
4.5
2.4
2.4
4.3
7.3
A test plane flies in a straight line with positive velocity v (t ), in miles per minute at a time t
minutes, where v is a differentiable function of t. Selected values of v (t ) for 0  t  40 are
shown in the table above.
a) Use a midpoint Riemann sum with four subintervals of equal length and values from the table
to approximate

40
0
v(t )dt. Show the computations that lead to your answer. Using correct units,
explain the meaning of

40
0
v(t )dt in terms of the plane’s flight.
b) Based on the values in the table, what is the smallest number of instances at which the
acceleration of the plane could equal zero on the open interval 0  t  40? Justify your answer.
 t 
 7t 
c) The function f, defined by f (t )  6  cos    3sin   , is used to model the velocity of
 10 
 40 
the plane, in miles per minute, for 0  t  40. According to this model, what is the acceleration
of the plane at t  23? Indicate units of measure.
d) According to the model f, given in part c), what is the average velocity of the plane, in miles
per minute, over the time interval 0  t  40 ?