Download Math 132. Practice Test 2 1. Practice on the new differentiation

Math 132. Practice Test 2
1. Practice on the new differentiation/integration formulas:
(a) Differentiate the following with respect to x:
(i) arcsin u
(ii) arctan u
(iii) arcsec u
(b) State the integral forms of the differentiation rules in (a)
(c) State the definitions of sinh x, cosh x and tanh x
(d) Differentiate sinh x, cosh x and tanh x. Then write the integral form of these differentiation
rules.
(e) Applications of some of the differentiation rules you stated above:
Differentiate: arctan(ex + sin x)
e2x cosh(5 − x3 )
(f) Applications of some of the integration rules you stated above:
Z
Z
Z
ex
Evaluate:
cosh(5x) dx
dx
x sinh(x2 ) dx
1 + e2x
5x2 arcsin(e4x )
Z
cosh(3x) dx
1 + sinh(3x)
(g) Use the definitions of sinh x and cosh x given above to show
(i) cosh2 x − sinh2 x = 1
(ii) sinh 2x = 2 sinh x cosh x
2.
A rectangular billboard 7 feet in height stands in a field so that its bottom is 8 feet
above the ground. A cow with eye level at 4 feet above the ground stands x feet from the
billboard, as illustrated in the diagram below.
(a) Express θ, the vertical angle subtended by the billboard at her eye, in terms of x.
(b) Find the distance the cow must stand from the billboard to maximize θ.
Z
3. Evaluate the integral
Z
4. Evaluate
√
sinh 7x
√
dx
7x
8x dx
√
2x2 − x4
5. An unknown radioactive element decays into non-radioactive substances. In 660 days the
radioactivity of a sample decreases by 68 percent.
(a) What is the half-life of the element?
(b) How long will it take for a sample of 100 mg to decay to 90 mg?
6. Solve the differential equation
dy
= 88xy 10 subject to y(0) = 4.
dx
7. Biologists stocked a lake with 233 fish and estimated the carrying capacity to be 9700.
The number of fish tripled in the first year. Assuming the fish population satisfies the logistic
L
, find P (t) and determine how long it will take for the population
equation P (t) =
1 + be−kt
to reach 4850.
8. Find the area between the curves x = 4 − y 2 and x = y − 2.
9. The region between the graphs of y = x2 and y = 2x is revolved around the line y = 4.
Find the volume of the resulting solid.
10. Find the volume of the solid obtained by revolving the the region bounded by the curves
y = 1/x3 , y = 0, x = 3 and x = 5 about the line y = 3.
11. Differentiate: (a) f (x) = ex arctan(x2 )
(b) g(x) = x arcsin x +
√
1 − x2