1151 Midterm 5 – Review Problems
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Problem 1 The figure shows aright trianglein the first quadrant. One side of the triangle is along the x -axis and the hypotenuse runs from the origin to a point on the parabola y = 4 x^2. Find the x and y coordinates that maximize the area of the triangle.
1 2 3
1
1
2
3
4
5
y = 4 x^2
( x,y )
Problem 2 Show your work and justify your answer.
(a)Find the point on the curve y =
x which is closest to the point(4 , 0).
(b)A cruise line offers a trip for$1000 per passenger. If at least 100 passenters sign up, the price is reduced
for all passengers by$5 for every additional passenger (beyond 100) who goes on the trip. The ship can
accommodate 250 passengers. What number of passenters maximizes the cruise lines total revenue?
What price does each passenger pay?
(c)Find the dimensions of the right circular cylinder of maximum volume that can be placed inside a
sphere of radius R.
(d)A certain tank consists of a cylinder with hemispherical ends. For a given surface area, describe the
shape and find the dimensions of the tank with maximum volume.
(e)A square piece of tin, 24 cm on each side, is to be made into a box with no top, by cutting a square
from each corner and bending up the resulting flaps to form the sides. How large a square should be
cut from each corner to make the volume of the box as large as possible?
Problem 3 Determine the form of the following limits, then evaluate them. Use lHopitals Rule where appropriate.
(a) lim
x 0 +
( ex x )
(^1) x (b) lim x 0 + (tan( x )) x 2 (c) lim x ln( x^10 ) x (d) lim x 0 ex 1 x x^2 (e) lim x ( 2 ) cos( x ) sin(2 x ) ( x 2
) 2
(f) lim x
(
x
x^2 + 4 x
)
(g) lim x
(
1 +
a
x
) x
for a a positive constant.
Problem 4 Evaluate the expressions:
(a)
^3
k =
(4 k 1) (b)
^3
k =
cos
(
k
6
)
Problem 5 Determine the following indefinite integrals. Check your work by differentiation.
(a)
(
e^3 x +1 4
x + 2
)
dx
(b)
4 + x^2
1 + x^2 dx
(c)
cos^3 ( ) + 1
cos^2 ( )
d
(d)
y (^2) 3 y y dy (e)
5 t ( t^2 3 t ) dt
(f)
x^2 3
x
(^3) x dx (g)
(
sec^2 ( ) + sec( ) tan( )
)
d
Problem 6 Suppose the function f is a solution to the initial value problem
dy
dx = e
x^2 10 xex^2
y (0) = 2
(a)Find the x -values of the critical points of f.
(b)Classify each critical point as a local minimum, local maximum, or neither.
(c) Estimate the value of f at its global maximum.
Problem 7 Given f ( x ) = 8 x x^2 on[0 , 8]with n = 4, complete the following steps:
Step 1: Calculate x and the grid pionts x 0 ,x 1 ,…,xn.
Step 2: Illustrate the right Riemann sum by sketching appropriate rectangles in the figure below.
1 2 3 4 5 6 7 8
5
10
15
20
y = f ( x )
Step 3: Calculate the right Riemann sum.
Problem 8 MULTIPLE CHOICE: Circle the correct answer in each part.
The figure illustrates the Riemann sum
n
k =
f ( x k ) x for a function f on the interval[4 , 8]. Use the figure to
answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
y = f ( x )
(a)Given the interval[4 , 8], what is the value of n?
(i) n = 3
(ii) n = 8
(iii) n = 5
(iv) n = 4
(v)None of the others.
(b)What is the value of x?
(i) x = 4
(ii) x = 0
(iii) x = 1
(iv) x = 2
(v) x =
4
k
(vi)None of the others.
(c)Given that x 0 = 4, find the expression for the grid points xk for k = 1 , 2 ,...,n.
(i) xk = 4 + ( k 1) x
(ii) xk = 4 + k x
(iii) xk = k
(iv) xk = k x
(v)None of the others.
(d)What is x k for k = 1 , 2 ,...,n?
(i) x k =
xk 1 + xk
2
(ii) x k = xk
(iii) x k = xk 1
(iv) x k = x 0
(v)None of the others.
(e)Evaluate this Riemann sum.
(i) f (5) + f (6) + f (7) + f (8)
(ii) f (6) + f (7) + f (8)
(iii) f (4) + f (5) + f (6) + f (7)
(iv)None of the others.
(f)This Riemann sum is a
(i)midpoint Riemann sum
(ii)right Riemann sum
(iii)left Riemann sum
(iv)None of the others.
Problem 9 MULTIPLE CHOICE: Circle the correct answer.
Consider the following limit of Riemann sums of a function g on an interval[ a,b ].
n lim
n
k =
x k cos( x k ) x on the interval[0 , ]
express this limit as a definite integral.
(a)
n
0
(
sin( x
(^2) ) 2
)
dx
(b)
0
(
sin( x^2 )
2
)
dx
(c)
0
x cos( x ) dx
(d)
n
0
x cos( x ) dx
(e)
x
0
t cos( t ) dt
(f) None of the others.
Problem 10 Suppose that f ( x ) 0 on the interval[1 , 3], that f ( x ) 0 on[3 , 5], that
3
1
f ( x ) dx = 4,
and that
5
3
f ( x ) dx = 9.
(a)Evaluate the following integrals.
(i)
5
1
3 f ( x ) dx (ii)
5
1
| f ( x )| dx (iii)
3
1
( f ( x ) x + 2) dx
(b)Assuming that f is odd , evaluate
3
5
f ( x ) dx.
(c)Assuming that f is even , evaluate
3
5
f ( x ) dx.
Problem 11 MULTIPLE CHOICE: Circle the correct answer in each part.
Given that
3
0
f ( x ) dx = 4, and that
6
3
f ( x ) dx = 6 , answer the following.
(a)Evaluate
0
3
5 f ( x ) dx.
(i) 20
(ii) 20
(iii) 4
(iv) 4
(v) 30
(vi) 30
(vii) 18
(viii) None of the others.
(b)Evaluate
6
0
f ( x ) dx.
(i) 4
(ii) 6
(iii) 20
(iv) 2
(v) 10
(vi)None of the others.
(c)Evaluate
6
3
( f ( x ) + 2) dx.
(i) 10
(ii) 4
(iii) 0
(iv) 12
(v) 8
(vi) 18
(vii)None of the others.
Problem 12 The acceleration function (in m/s^2 ), the velocity, and the position at t = 0are given for a particle moving along a line.
a ( t ) = 2 t sin
(
4
t
)
, for 0 t 8
v (0) = 3
s (0) = 4
Find:
(a)The velocity function, v ( t ).
(b)The position function, s ( t ).
(c)Using the position function you found, s ( t ), find the net displacement between t = 0and t = 8.
Problem 13 The acceleration function (in m/s^2 ), the velocity, and the position at t = 0are given for a particle moving along a line.
a ( t ) = 2 t 4 , for 0 t
v (0) = 3
s (0) = 0
Find:
(a)The velocity function, v ( t ).
(b)The position function, s ( t ).
(c)Graph both the velocity and position functions.
Problem 14 Graph several functions that satisfy the differential equation f ( x ) = 3 x^2 1. Then, find and graph the particular solution that satisfies the initial condition f (2) = 1.
Problem 15 (a)The graph of a function g is shown in the figure. Use geometry to evaluate the integral 9 1
g ( x ) dx.
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
y = g ( x )
(1 , 2)
(5 , 6)
(9 , 2)
(b)The graph of a function f is given in the figure.
1 2 3 4 5 6
2
1
1
2
y = f ( t )
1 / 4 of circle of radius 2
(i)Compute
6
0
f ( t ) dt.
(ii)Compute
6
0
| f ( t )| dt.
(c)The graph of a function h is shown in the figure below.
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
y = h ( x )
(1 , 2)
(8 , 4)
(9 , 2)
(i)Use geometry to evaluate the integral
9
1
h ( x ) dx.
(ii)Draw a rectangle, whose base is along the interval[1 , 9]along the x -axis, and whose net area is
equal to
9
1
h ( x ) dx.
(d)The graph of a function f is shown in the figure below.
1 2 3 4 5 6 7
1
0. 5
0. 5
1
y = f ( t )
(i)Compute
7
0
f ( t ) dt.
(ii)Compute the geometric area of the region between the graph of the function f and the hori- zontal axis on the interval[0 , 7].