MATH 105 PRACTICE MIDTERM III KEY
Please work out each of the given problems. Credit will be based on the steps that you show towards the final answer. Show your work.
Please answer the following true or false.
If false, explain why or provide a counter example.
If true, explain why.
A) If f(x) is a positive continuous function such that
equal to 3.
True, since f is a positive function, both integrals represent areas, and the second integral represents an area of a region that contains the region of the first integral. Hence the second integral must be at least as large as 4.
B) If f(x) is a differentiable function such that the equation of the tangent line at
x = 2
and if x = 2 is the first guess in
Newton’s method, then x = 1
is the second guess.
True, We arrive at the second guess by finding the x-intercept of the tangent line. Since the x-intercept is 1, x = 1 is the second guess.
If f(x) and g(x)
are continuous functions on [a,b],
False, for example, if f(x) = g(x) = 1 and if a = 0 and b = 2, then the left hand side is 2 and the right hand side is (2)(2) = 4
PROBLEM 2 Evaluate the
Just integrate the terms individually:
-cos x - 2/3 x3/2 + 2/3 x3 - 3x + C
We use u-substitution:
u = 1 - x du = -1dx dx = -du x = 1 - u
x = 2 u = -1
x = 3 u = -2
We use u-substitution:
u = 2x du = 2dx dx = 1/2 du
Substituting, we get
-1/2csc(2x) + C
PROBLEM 3 Use Riemann
Sums to find the area of the region below the curve
y = 9 - x2, above the x-axis,
and between x = 1 and x = 3.
Dx = (3 - 1) / n = 2/n
Use the chain rule along with the second fundamental
theorem of calculus:
Use the chain rule along with the second fundamental theorem of calculus:
u = sin x
u'(x) = cos x
F'(u) = cos u2 = cos(sin2 x) By the fundamental theorem of calculus
Now use the chain rule
F'(x) = u'(x)F'(u) = (cos x)(cos(sin2 x))
You are the
owner of Tahoe Winter Wear and need to determine the best price to sell your
most popular winter jacket. Your
cost for selling the jackets is
= 50 + 20x
is the amount to jackets that you sell.
Your research shows that the relationship between price, p,
and the number of jackets that you can sell, x, is
First calculate the revenue R:
R = xp = x(300 - 10x) = 300x - 10x2
Now use the fact that Profit equals Revenue minus Cost:
P = R - C = (300x - 10x2) - (50 + 20x) = 280x - 10x2 - 50
To find the maximum profit we take the derivative and set it equal to zero:
P' = 280 - 20x = 0
x = 14
Now substitute 14 for x in the demand equation:
p = 300 - 10(14) = 160
You should charge $160 for the jacket.
manufacturing a square computer chip. Your
machine can construct the square with side length 0.4
Use differentials to approximate the maximum percent error in the area of
Use the formula
A = x2
A' = 2x
DA @ 2xDx = 2(0.4)(0.0002) = 0.00016
To find the percent error, use
Percent Error = (DA/A)(100%) = (0.00016/.42)(100%) = 0.1%
x0 = 1 x1 = 1.5 x2 = 2 x3 = 2.5 x4 = 3
f(1) = 3.9 f(1.5) = 3.7 f(2) = 3.5 f(2.5) = 3.1 f(3) = 2.6
Dx = (3 - 1)/8 = 0.25
Now use the trapezoid formula:
0.25[3.9 + 2(3.7) + 2(3.5) + 2(3.1) + 2.6] = 6.775