Section 5.1 (Part I)
1) 5.1.2 Verify the statement Int(3/sqrt(x) dx) = 6sqrt(x) + C by differentiating the right side. ( Int(f(x) dx) is the integral of f(x) with respect to x.)
2) 5.1.4 Verify the statement Int(2 - 1/((x^3)^(1/4)) dx) = 2x - 4x^(1/4) + C by differentiating the right side.
3) 5.1.6 Verify the statement Int( 4sqrt(x)(x^2-2) dx) = (8x^(3/2)(3x^2-14))/21 + C by differentiating the right side.
4) 5.1.8 Verify the statement Int((2x-1)/x^(4/3) dx) = 3(x+1)/x^(1/3) + C by differentiating the right side.
5) 5.1.12 Find Int(3t^4 dt) and check your answer by differentiation..x
6) 5.1.14 Find Int(-5y^-4 dy) and check your answer by differentiation..
7) 5.1.20 Find Int(v^(-1/2) dv) and check your answer by differentiation.. x
8) 5.1.30 Find two functions which have the derivative f '(x) = -1. Sketch and describe these functions.
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Section 5.1 (Part II)
36, 37, 40, 44, 46, 50, 56, 60, 63, 64, 68, 71, 76
1) 5.1.36 Find Int((5sqrt(x) + 1/(3sqrt(x) ) dx) and check your answer by differentiation.
2) 5.1.40 Find Int(1/(4x^2) dx) and check your answer by differentiation.
3) 5.1.44 Find Int(sqrt(x)(x+3) dx).
4) 5.1.46 Find Int((t-5)^2 dt).
5) 5.1.50 Find y = f(x) when f '(x) = 1/5x - 2 and f(10) = -10.
6) 5.1.56 Find y when dy/dx = 2(x-1) and y(3) = 1.
7) 5.1.60 Find f(x) when f ''(x) = x^2, f '(0) = 6 and f(0) = 3.
8) 5.1.64 If the marginal cost dC/dt = 1/30x + 30 and C(0) = $1500. Find C(x).
9) 5.1.68 If the marginal revenue function dR/dx - 275 - 3x and knowing R(0) = 0, find the revenue and demand functions.
10) 5.1.76 Given the population growth function dP/dt = 500t^1.06, and that the current population is P=50K, find P in 10 yrs.
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Section 5.2 (Part I)
1) 5.2.2 Identify the u and du/dx you would use to put the integral Int((7-5x^2)^4(-10x) dx) into the form Int(u^n (du/dx) dx).
2) 5.2.4 Identify the u and du/dx you would use to put the integral Int(4x^3 * sqrt(x^4+1) dx) into the form Int(u^n (du/dx) dx).
3) 5.2.6 Identify the u and du/dx you would use to put the integral Int( 5/((3+5x)^2)dx) into the form Int(u^n (du/dx) dx).
4) 5.3.8 Identify the u and du/dx you would use to put the integral Int((4 - sqrt(x))^2*(-1/(2sqrt(x))) dx) into the form Int(u^n (du/dx) dx).
5) 5.2.12 Find the indefinite integral Int(sqrt(3-x^3)*(3x^2) dx) and check your answer through differentiation.
6) 5.2.18 Find the indefinite integral Int(x^2/(x^3-1)^2 dx) and check your answer through differentiation.
7) 5.2.22 Find the indefinite integral Int((6x^2+6)/(x^3 + 3x + 9)^4 dx) and check your answer through differentiation.
8) 5.2.26 Find the indefinite integral Int(x^2/sqrt(1-x^3) dx) and check your answer through differentiation.
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Section 5.2 (Part II)
1) 5.2.30 Find the indefinite integral Int(5x/sqrt(3-10x^2) dx) and check your answer through differentiation.
2) 5.2.36 Use formal substitution to find the indefinite integral Int(x^2(1-x^3)^2 dx).
3) 5.2.42 Use formal substitution to find the indefinite integral Int(sqrt(x)*(7-2x^(3/2))^(1/3) dx).
4) 5.2.54 Find the demand fuction x = f(p) when dx/dp = -400/(102p-1)^3 and x = 10000 when p = $100.
5) 5.2.58 Find the income consumed Q in terms of income x when dQ/dx = 0.94/(x-19999)^(0.06) where x >= 20000. Find the consumed income and income saved (x - Q) when the income is $75000 and $125000.
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Section 5.3
1) 5.3.4 Using the Exponential Rule, find the indefinite integral Int(e^(-.25x) dx).
2) 5.3.10 Using the Exponential Rule, find the indefinite integral Int(3(x-4)e^(x^2-8x) dx).
3) 5.3.16 Using the Log Rule, find the indefinite integral Int(1/(6x-5) dx).
4) 5.3.20 Using the Log Rule, find the indefinite integral Int(x^3/(5+x^4) dx).
5) 5.3.22 Using the Log Rule, find the indefinite integral Int(x/(x^2+4) dx).
6) 5.3.24 Using the Log Rule, find the indefinite integral Int((3x^2 + 6x + 9)/(2x^3 + 6x^2 + 18x + 1) dx).
7) 5.3.28 Using the Log Rule, find the indefinite integral Int(e^x/(1+e^x) dx).
8) 5.3.36 Find the indefinite integral Int(xe^(x^2)/(2+e^(x^2) dx).
9) 5.3.46 Find the indefinite integral Int((6x+e^x)*sqrt(3x^2+e^x) dx). x
10) 5.3.48 Find the indefinite integral Int((x-1)/5x dx).
11) 5.3.58 Find population P in terms of time(in days) t when dP/dt = -125e^(-t/20) and when t = 0, P = 2500. What is the population after 15 days? How long will it take for the entire population to die?
12) 5.3.60 Marginal revenue of the sale of the product can be modeled by dR/dx = 40 - .03x + 100/(x+2). Find the revenue function. Find the revenue when 1500 units are sold find the number of units sold when the revenue. Find the number of units sold when the revenue is $27386.50.
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Section 5.4
1) 5.4.4 Sketch the region of the area represented by the integral Int((2x + 1) dx, 0, 3) and use a geometric formula to evaluate the integral. (Note: Int(f(x) dx, a, b is the integral of f(x) with respect to x from a to b.)
2) 5.4.7 Sketch the region of the area represented by the integral Int(sqrt(9-x^2) dx, -3, 3) and use a geometric formula to evaluate the integral.
3) 5.4.10 Given that Int(f(x) dx, 0, 5) = 8 and Int(g(x) dx, 0, 5) = 3 find the following integrals. Int(2g(x) dx, 0, 5), Int(f(x) dx, 5, 0), Int(f(x) dx, 5, 5), and Int((f(x)-f(x)) dx, 0, 5)
4) 5.4.17 Evaluate the definite integral Int( (x^2 + 4)/x dx, 1,4).
5) Evaluate the definite integral Int((3x^2 + x -2) dx, 0, 3) x
6) 5.4.20 Evaluate the definite integral Int(3v dv, 4, 9).
7) 5.4.28 Evaluate the definite integral Int(sqrt(2/x) dx, 1,4). x
8) 5.4.30 Evaluate the definite integral Int((2x-sqrt(x))/7 dx, 0,1).
9) 5.4.46 Evaluate the definite integral Int((3-|x|) dx, -3, 3).
10) 5.4.50 Evaluate the definite integral Int((3-ln x)^4/2x dx, 1,2).
11) 5.4.52 Evaluate the definite integral Int((x-3) dx, 0, 3).
12) 5.4.56 Evaluate the definite integral Int((e^x)/2 dx, 0, ln 8).
13) 5.4.60 Find the area bounded by the graphs y = 2e^x, y = 2, x = 1, x = 4.
14) 5.4.63 Find the average value of the function 5e^(0.2*(x-10)) on the interval [0, 10].
15) 5.4.66 Find the average value of the function 1/(x-3)^2 on the interval [0,2]. x
16) 5.4.70 State whether the function 3x^5 + 2x is even, odd, or neither.
17) 5.4.76 Find the change in cost C from x = 10 to x = 13 when dC/dx = 15000/x^3.
18) 5.4.78 Find the change in revenue R from x = 500 to x = 505 when dR.dx = 50(21 + 800/x).
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Section 5.5 (Part I)
1) 5.5.6 Find the area of the region bounded by f(x) = (x-1)^3 and g(x) = x-1 on [0,2]. x
2) 5.5.10 Sketch and describe the region whose area is described by the integral Int(((1-x^2) - (x^2 - 1)) dx, -1, 1).
3) 5.5.20 Sketch and describe the region bounded by y = 16 - x^4 and y = x^4. Find the area of this region.
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Section 5.5 (Part II)
23, 28, 32, 37, 41, 44, 49, 52, 57, 58
1) 5.5.23 Sketch and describe the region bounded by y = 8/x, y = x^2, x = 1 and x = 4. Find the area of the bounded region. x
2) 5.5.28 Sketch and describe the region bounded by f(y) = y(3 - 2y), g(y) = -5y. Find the area of the bounded region.
3) 5.5.32 Write the integral needed to evaluate the area of the region bounded by f(x) = 3x(x^2 - 3x - 2), g(x) = x^2.
4) 5.5.44 Find the consumer and producer surpluses when p1(x) = 1000 - .4x^2 is the demand and p2(x) = 42x is the supply.
5) 5.5.52 An epidemic was spreading such that t weeks after the initial outbreak it had infected N1(t) = 0.2t^2 + .4t + 150 for 0 <= t <= 50. 30 weeks after the initial outbreak a vaccine was developed and then the number of people infected was described by the model N2(t) = -0.4t^2 + 7t + 200. Find the number of people that the vaccine prevented from becoming ill during the epidemic.
6) 5.5.58 Letting x represent the percent of families in a nation and y the precent of total income. The model y = x assumes that each family in a country has the same income and the model y = f(x) gives the actual income distribution. The area between these two curves gives the "income inequality". For the United States the Lorenz curve for the year 2001 can be modeled by y = (0.00059x^2 + 0.0233x + 1.731)^2 where x is in the interval [0,100]. Find the income inequality for the United States during this year.