Problems for Assignments 1-3

Calculus 2

Section 6.1

1) *6.1.2 Sketch and describe two functions F such that F' = f when f is the line going through (1,0) and (0,1).

2) *6.1.4 Sketch and describe two functions F such that F' = f when f(x) = |x|

3) 6.1.6 Estimate f(x) for x = 2,4,6 when f(0) = 50 and f '(0)= 17, f '(2)= 15, f '(4) =10, and f '(6) = 2.

4) *6.1.8 Let dP/dt be described by the piecewise defined function which follows. When t <= 2, dP/dt = -1. When 2 <= t <= 4, dP/dt = t - 3. When t >= 4, dP/dt = 1. Sketch dP/dt and using P(0) = 2 use dP/dt to find P(t) when t = 1,2,3,4,5.

5) f(x) is a function defined on the interval 0 <= x <= 7 with the following characteristics:

f '(x) = 1 for all x on the interval (0,2),
f '(x) = -1 for all x on the interval (2,3),
f '(x) = 2 for all x on the interval (3,4),
f '(x) = -2 for all x on the interval (4,6),
f '(x) = 1 for all x on the interval (6,7)
f(3) = 0

What is the integral of f ' (x) over the interval 0 <= x <= 7?

Describe your graph of f(x), indicating where it is increasing and decreasing and where it is concave up, where it is straight and where it is concave down.

What are the following values:

f(2)
f(3)
f(4)
f(6)
f(7)

Is the graph of f(x) continuous?

6) *6.1.10 Let f (x)= x^3 - 3x^2 and F'(x) = f(x). Sketch or graph f(x) (with a graphing calculator if necessary, though you should be able to construct the graph easily) and referring only to the graph of f(x), give the critical points, explain which are local maxima/minima, and which are neither. Also sketch a possible graph of F(x).

7) *6.1.12 Sketch and describe two functions F with F'(x) = f(x) (Sketch or graph f(x) with a graphing calculator) by only looking at the graph of f(x) = x^3 - 4x^2 + x + 6. In one let F(0) = 0 and the other F(0) = 1. Also identify local minima, maxima, and points of inflection.

8) *6.1.18 Let g' ={-x when 0 <= x <= 10, 4x when 10 <= x <= 15, x when 15 <= x <= 20, and -(1/2)x when x >= 20}. Sketch g' and given g(0) = 50 sketch the graph of g(x) and specify all critical and inflection points of g(x).

9) 6.1.22 Use a graph of 4cos(x^2) to determine where an antiderivative, F, of this function reaches its maximum on 0<= x<= 3. If F(1) =4, find this maximum obtained.

10) Given the graph below, which depicts outflow vs. clock time and inflow vs. clock time for a reservoir:

When was the quantity of water greatest and when least? Describe in terms of the behavior of the two curves.
When was the quantity of water increasing fastest, and when most slowly? Describe in terms of the behavior of the two curves.

*** See the text section for this graph. In the current edition the problem is number 25; in other editions it shouldn't be far from that. ***

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Section 6.2

1) Find an antiderivative of f(x) = 4x.

2) Find an antiderivative of r(t) = -1/(t^2).

3) Find an antiderivative of f(t) = 3t^3 + 4t^4 -t^2.

4) Find the general antiderivative of f(z) = 3z + 2e^z.

5) Find an antiderivative F(x) with F'(x) = f(x) and F(0) = 0. f(x) = x^2. Is this the only possible solution and why?

6) Find an antiderivative F(x) with F'(x) = f(x) and F(0) = 0. f(x) = 2 + 3sqrt(x). Is this the only possible solution and why?

7) Find the indefinite integral Int(5/t^3 dt). (Where Int(f(t) dt) is the integral of f(t) with respect to t.)

8) Find the indefinite integral of t `sqrt(t) + 1 / (t `sqrt(t)).

9) Find the definite integral of sin(t) + cos(t), on the interval 0 <= t <= `pi/4.

10) Find the indefinite integral Int(-1/(e^z) dz).

11) Find the definite integral Int(x^3- x/2 dx, 1, 4) (Where Int(f(x) dx, a, b) is the integral of f(x) from a to b with respect to x)

12) Calculate the exact area between the x-axis and the graph of y = 3x^2 + x - 4

13) The average value of the function 6/x^2 is 1 on the interval [1,c]. Find c.

14) Find the indefinite integral of e^(5+x) + e^(5x)

15) The origin and the point (a,a) are at opposite corners of a square. Calculate the ratio of the two parts of the square into which the curve y = (x^2)/a divides the square.

16) In drilling an oil well, the total cost C, consists of fixed costs (which do not change with respect to depth) and marginal costs (which change with respect to depth). Drilling becomes more expensive as the well goes deeper into the earth. Suppose the fixed costs are 250,000 dollars and the marginal costs are C ' (x) = 1000 + 3x dollars/meter, where x is the depth in meters. Find the total cost of drilling a well x meters deep.

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Section 6.3

1) 6.3.2 Find the general solution of the differential equation dy/dx = 9x - 2/x.

2) 6.3.4 Find the general solution of the differential equation dr/dp = 4 cos p.

3) 6.3.6 Find the solution of the initial value problem: dy/dx = 8x^3 +2x, y(2) = 10.

4) 6.3.8 Find the solution of the initial value problem: ds/dt = -32t + 100, s = 50 when t = 0.

5) 6.3.10 Find the general solution of the differential equation dy/dx = 2x + 1. Graph three different solutions of this equation.

6) 6.3.16 A water balloon is launched from the roof of a building at time t = 0 and velocity v(t) = -32t + 40ft/s at time t. v > 0 corresponds to vertical motion. If the roof of the building is 30 feet above the ground, find an expression for the height of the water balloon above the ground at time t. What is the average velocity of the balloon between t = 1.5 and t = 3 seconds? For a 6-foot person is standing on the ground below, how fast is the balloon when it hits them in the head.

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Section 6.4

1) 6.4.6 Assume that F'(t) = sin t * cos t and F(0) =1. Find F(b) for b = 0, 1/2, 1, 3/2, 2, 5/2, 3.

2) 6.4.8 Write an expression for the function f(x) with the given properties. f'(x) = (sin x)/x and f(1) = 5.

3) 6.4.12 Find d/dx(Int(ln(t) dt, x, 1)).

Explanation of notation: int( f(x) dx, a, b) means the integral of f(x) with respect to x, from x = a to x = b.

4) 6.4.18 Find d/dx(Int(e^-(t^2) dt, 0 , x^3)).

5) 6.4.22 Let F(x) = Int(sin(4t) dt, 0, x). Evaluate F(pi). Draw a sketch and describe geometrically why your value for F(pi) is correct. For what values of x is F(x) positive and for which value is it negative?

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Section 6.5

1) 6.5.2 At time t = 0, a stone is thrown off a 250 meter cliff with velocity 15 m/s downward. Express its height, h(t), in meters above the ground as a function of time, t, in seconds.

2) 6.5.6 On the moon the acceleration due to gravity is 5ft/s^2. An astronaut jumps into the air with an initial upward velocity of 9ft/s. How high does he go and how long is the astronaut off the ground?

3) 6.5.8 Galileo stated: "The time in which any space is transversed by a body starting at rest and uniformly accelerated is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is the mean of the highest velocity and the velocity just before acceleration began."

Write this statement in symbols defining all of the symbols you use. Check this statement for a body dropping off a 100-ft building accelerating from rest under gravity until it hits the ground. Show why Galileo's statement is true in general.

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Section 7.1

1) 7.1.6 Find the integral Int( t sin(t^2) dt).

2) 7.1.10 Find the integral Int(y(y^2+7)^5 dy).

3) 7.1.12 Find the integral Int((x^3)(1+4x^4)^2 dx).

4) 7.1.18 Find the integral Int(sqrt(cos(3t))*sin(3t) dt).

5) 7.1.20 Find the integral Int((x^3)(e^(x^4+1)) dx).

6) 7.1.21 Find the integral Int((x^2)(e^(x^3+1)) dx).

7) 7.1.24 Find the integral Int(sin(a)(cos^2)(a) da).

8) 7.1.30 Find the integral Int(sin(sqrt(x))/sqrt(x) dx).

9) 7.1.35 Find the integral Int((t+1)^2/(t^2) dt).

10) 7.1.36 Find the integral Int((e^x + e^-x)/(e^x - e^-x) dx).

11) 7.1.40 Find the integral Int(cosh 3t dt).

12) 7.1.42 Find the integral Int(sinh(4w - 3) dw).

13) 7.1.60 Find the definite integral Int(1/(t+7)^2 dt, 1, 3).

14) Find the definite integral Int( sin x ( cos x + 5) ^7 dx, 0, pi)

15) Are the two integrals, Int((cos x)e^(sinx) dx) and Int((e^arcsin x)/sqrt(1-x^2)) the same?

16) Assume world population to be modeled by the function

World population P(t) = 5.3 e^(0.014 t)

where t is the clock time in years, as measured relative to Jan 1, 1990.

What were the populations in 1990 and 2000?
What was the average population between during the 1990's and how did you find it?

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Section 7.2

1) 7.2.6 Find the integral Int(xe^(-2x) dx).

2) 7.2.10 Find the integral Int(q^4 ln(4q) dq).

3) 7.2.12 Find the integral Int(sin^2(x) dx).

4) 7.2.16 Find the integral Int((t+2)sqrt(2+3t) dt).

5) 7.2.18 Find the integral Int(z^2/e^z dz).

6) 7.2.20 Find the integral Int(3y/sqrt(4+y) dy).

7) 7.2.24 Find the integral Int(arctan 4z dz).

8) 7.2.27 Find the integral Int(x^5 cos(x^3) dx).

9) 7.2.30 Evaluate the definite integral Int(ln t dt, 2, 4).

10) 7.2.36 Evaluate the definite integral Int(arccos z dz, -1, 1).

11) 7.2.40 In number 3) you found Int(sin^2(x) dx) using integration by parts most likely, if not then do so. Redo this integral using the identity sin^2(x) = (1 - cos(2x))/2. Explain any differences in the form of the solution found by the two methods.

12) 7.2.42 Find the integral Int(e^x(sin x) dx) (Do integration by parts twice and go from there).

13) 7.2.50 Given f(0) = 6, f(1) = 5, f '(1)=2. Find Int(x * f '(x), x, 0, 1).

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Section 7.3

1) 7.3.6 Find the integral Int(sin^3 (w) cos x dw) using integration tables. Mention which formula you use.

2) 7.3.12 Find the integral Int(1/sqrt(16x^2 + 9) dx) using integration tables. Mention which formula you use.

3) 7.3.15 Find the integral Int(x^4 e^(3x) dx) using integration tables. Mention which formula you use.

4) 7.3.18 Find the integral Int(u^3 ln(3u) du) using integration tables. Mention which formula you use.

5) 7.3.24 Find the integral Int(e^(3x)(cos 5x) dx) using integration tables. Mention which formula you use.

6) 7.3.30 Find the integral Int(tan^5(x) dx) using integration tables. Mention which formula you use.

7) 7.3.33 Find the integral Int(1/(1 + (z+2)^2) dz) using integration tables. Mention which formula you use.

8) 7.3.36 Find the integral Int(sin^5(x) dx) using integration tables. Mention which formula you use.

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Section 7.4

1) 7.4.6 Integrate the function 2y/(y^3 - y^2 + y - 1) with respect to y.

2) 7.4.8 Find Int(15/(16-x^2) dx).

3) 7.4.10 Find Int(9/(y^3 - 9y) dy).

4) 7.4.16 Find Int(2/(x^4-x^3) dx) using 2/(x^4-x^3) = A/x + B/x^2 + C/x^3 + D/(x-1).

5) 7.4.24 Complete the square and give a substitution which can used to compute Int(1/(x^2 + 2x + 2) dx).

6) 7.4.28 Find Int((y+2)/(2y^2 + 3y + 1) dy).

7) 7.4.29 Find Int((z-1)/sqrt(2z-z^2) dz).

8) 7.4.34 Find Int(1/((x+3)(x-5)) dx).

9) 7.4.36 Find the partial fraction decomposition of 1/( x * (L-x) ).

10) 7.4.38 Find Int(1/(5P^2-5P^3) dP).

11) 7.4.44 Find Int(3y^2/(9+y^2) dy).

12) 7.4.49 Integrate (z - 1) / sqrt( 2 z - z^2) with respect to z.

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Section 7.5

1) 7.5.10 Calculate LEFT(2), RIGHT(2), TRAP(2), and MID(2) for Int(sin x dx, 0, pi).

2) 7.5.13 Let f(x) be a function which is decreasing and concave down on [a,b] and consider Int( f(x) dx, a, b).

List the approximations {LEFT,RIGHT,TRAP,MID} in increasing order.
Where does the actual value of the integral lie in this set?

3) 7.5.14 Let f(x) be a function which in decreasing and concave up on [a,b] and conside Int(f(x) dx, a, b).

List the approximations {LEFT, RIGHT, TRAP, MID} in increasing order.
Where does the actual value of the integral lie in this set?

4) Let f be a positive, decreasing, concave up function on the interval [0,h]

Why is the area of the trapezoid h(L1 + L2)/2?
Describe the sketch of the area E = h* f(0).
Describe the sketch of the area F = h* f(h).
Describe the sketch of the area R = h* f(h/2).
Describe the sketch of the area C = h* [f(0) + f(h)]/2.
Describe the sketch of the area N = h/2* [f(0) + f(h/2)]/2 + h/2* [f(h/2) + f(h)].
Why is C= (E + F)/2?
Why is N= (R + C)/2?
Is E or F a better aprroximation and why?
Is R or C a better approximation and why?

5) 7.5.18 Let f(x) be a function which is concave up on [0,5] and has a relative minimum at x= 2.5. Which approximations will be overestimates and which will be underestimates. Which ones can you not tell with the information given if any?

6) 7.5.22 Show TRAP(n) = LEFT(n) + 1/2(f(h) + f(0))* `dx.

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Section 7.6

1) 7.6.1 Estimate Int(x^2 dx,0,6) using SIMP(2).

2) 7.6.3 Given the integral Int(1/x dx, 1, 2)

Using n = 2,4,8,16,32,64,128 find the left and right approximations and the corresponding errors.
What are the signs of the erros in the left and right approximations? How do the errors change when n is doubled?
For the same values of n, compute the midpoint and trapezoidal approximations and corresponding errors.
What are the signs of the errors for the midpoint and trapezoid approximations? How do the errors change if n is doubled?
For n = 2,4,8,16,32 compute Simpson's rule approx. and the error for each n. How do these errors change as you double n?

3) 7.6.6 The approximation of a definite integral using n = 10 is 2.346 and the exact value is 4.0. If the approximation was found using each of the following rules , use the same rule to estimate the integral with n = 30. LEFT, TRAP, and SIMP.

4) Let a < m < b, with m the midpoint between a and b.

If f(x) = x^2, show that int(f(x), x, a, b) = h/3 ( f(a)/2 + 2 f(m) + f(b) / 2).

If f(x) is amy quadratic function, show that int(f(x), x, a, b) = h/3 ( f(a)/2 + 2 f(m) + f(b) / 2).

Explain how this proves that Simpson's Rule gives the exact value of the integral when f(x) is a quadratic function.

5) If TRAP(10) = 12.676 and TRAP(30) = 10.420, estimate the actual value of the integral.

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Section 7.7

1) 7.7.6 Calculate the integral Int(1/(x+4)^2 dx, 3, inf) if it converges. (Note: inf here means infinity)

2) 7.7.12 Calculate the integral Int(1/(u^2 - 16) dx, 0, 4) if it converges.

3) 7.7.24 Calculate the integral Int(3/(x ln x) dx, 1 2) if it converges.

4) 7.7.30 Calculate the integral Int(100/sqrt(y-6) dy, 7, inf) if it converges.

5) 7.7.36 For what values of p does the integral Int(x^p ln x dx, 0, e) converge? What is the value of the integral when it does converge?

6) 7.7.40 The kth moment mk of the normal distribution is given by mk = 1/sqrt(2pi) * Int( x^k * e ^ (-x^2/2) dx, -inf, inf). Use the fact that Int(e^(-x^2/2) dx, -inf, inf) = sqrt(2pi) to find m2 and m4.

7) 7.7.44 The rate, r, at which people get sick during an epidemic of the flu can be modeled by r = 1000te^(-0.5t) where r is people/day and t is days since start of epidemic.

Sketch and describe the graph of r as a function of t.
At what time are people getting sick the fastest?
How many people get sick altogether?

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Section 7.8

1) 7.8.6 Does the integral Int((3x^2 - 6x + 1)/(4x^2 + 9) dx, 1, inf) diverge or converge and why?

2) 7.8.12 Does the integral Int( 1/(x^4 + 2) dx, 1, inf) diverge or converge and why?

3) Does the integral Int(1/sqrt(x^2+1) dx, 1, inf) diverge or converge and why?

4) 7.8.20 Does the integral Int(1/sqrt(x^3 + x) dx, 0, 1) diverge or converge and why?

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Section 8.1

1) 8.1.4 Consider a circle with radius 3, and a horizontal strip with width `dy at position y above the origin.

Write a Riemann sum for the reigon
Write a definite integral representing the area of the reigon.
Evaluate this integral exactly.

2) 8.1.6 Consider the area between the line y = h and the graph of y = |x| containing a horizontal strip with width `dh.

Write a Riemann sum for the reigon
Write a definite integral representing the area of the reigon.
Evaluate this integral exactly.

3) 8.1.12 Consider the half disk with radius 7m and thickness 10m. `dy is parallel to the base.

Write a Riemann sum for the reigon
Write a definite integral representing the area of the reigon.
Evaluate this integral exactly.

4) 8.1.18 Consider the integral Int(7(1-h/3) dh, 0, 4).

What shape does this integral represent?
If it is a triangle what is its height or if it is a circle what is its radius?
Make and describe a detailed sketch of the reigon which shows the variable and other relevent quantities.

5) 8.1.20 Consider the integral Int(pi(x/4)^2 dx, 0, 10).

What shape does this integral represent?
If it is a cone what is its height and radius or if it is a sphere what is its radius?
Make and describe a detailed sketch of the reigon which shows the variable and other relevent quantities.

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Section 8.2

1, 3, 6, 8, 11, 13, 16, 19, 20, 23, 26, 27, 31, 34, 37, 40, 41

1) 8.2.6 Find the volume of the reigon bounded by y = sin x, y = 0, x = 0, x = pi

2) 8.2.8 Find the volume of the reigon bounded by sqrt(cosh 2x) y = 0, x = 0, x = 2 rotated around the x-axis.

3) 8.2.11 Find the arc length of the graph of y = sqrt(x^3) from x = 0 to x = 2.

4) 8.2.16 Find the length of the parametric curves, x = sin(2t), y = cos(3t) for 0 <= t <= 2pi.

5) 8.2.20 Sketch the solid obtained by rotation the reigon y = sqrt(x), x = 4, y = 0 rotated around the line x = 4. Using your sketch describe how to approximate the volume of the solid by a Riemann sum and then find the volume.

6) 8.2.26 Consider the region of the solid bounded by y = x^3, y = 1, and the y-axis. Find the area of the solid whose base is this reigon and whose cross-sections perpendicular to the x-axis are circles

7) 8.2.31 Consider the region bounded by y= e^x, the x-axis, and the lines x = 0 and x = 1. Find the area of the solid whose base is this reigon and the cross-sections perpendicular to the x-axis are squares.

8) 8.2.34 A 1m gutter is made of three strips of metal, each 7cm wide. The sides of the gutter form a trapezoid without a base where the angles between the pieces of metal are 120 degrees.

Find the volume of water in the gutter when the depth of the water is h cm.
What is the maximum value of h?
What is the maximum volume of water the gutter can hold?
If the gutter is holding half of the maximum amount of water, is the depth larger or smaller to the maximal h you found earlier?
Find the depth of the water when the gutter is holding half of the maximum amount of water.

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Section 8.3

1) 8.3.6 Convert the point (1,0) from Cartesian coordinates to polar coordinates.

2) 8.3.12 Graph the equation r = 1 - n sin(theta) for n = 2,3,4. What is the relationship between n and the shape of the graph.

3) 8.3.18 Give an inequality describing the reigon inside the sector of a circle of radius 2 from (sqrt(3), 1) to (sqrt(3),-1).

4) 8.3.24 In polar coordinates write the quations for the line x = 2 and the circle of radius 3 centered at the origin.

Write an integral for the area of the reigon to the right of x = 2 and inside the circle.
Evaluate the integral.

5) 8.3.30 Sketch the bounded reigon inside the lemnisate r^2 = 4 cos 2(theta) and outside the circle r = sqrt(2). Evaluate the area which you have sketched.

6) 8.3.36 Find the arc length of the curve 2(theta), 0 <= theta <= 2pi.

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Section 8.4

1) 8.4.4 If a rod lies along the x-axis between a and b, the moment of the rod is given by Int(x rho(x) dx,a,b), where rho(x) is the density in g/m in terms of x. Find the moment of a 2m rod whose density is given by rho(x) = 2 + 6x g/m.

2) 8.4.6 Find a Riemann sum which approximates the total mass of a 4 x 6 rectangular sheet whose density per unit area at a distance x from one side of length 6 is 1/(1 + x^5). Then calculate the actual mass of the sheet.

3) 8.4.12 Consider the reigon on the interval [a,b] which is bounded above by f(x) and below by g(x). If the density rho only varies with x then find expressions for the total mass in terms of f(x), g(x) and rho(x).

4) 8.4.18 A rod with length 3m and density rho(x) = 2 + x^3 g/m is positioned along the positive x-axis with its left end at the origin. Find the total mass and the center of mass of this rod.

5) 8.4.24 A metal plate, with constant density 5g/cm^2, has a shape bounded by the curve y = sqrt(2x) and the x-axis with 0 <= x <= 1.

Find the total mass of the plate.
Find x_bar and y_bar where (x_bar, y_bar) is the center of mass.

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Section 8.5

1) 8.5.6 A worker on a scaffolding 65ft above the ground needs to lift a 400lb bucket of cement from the ground to a point 30ft above the ground by pulling on a rope weighing 0.75 lb/ft. How much work is required?

2) 8.5.10 A rectangular water tank has length 15ft, width 15ft, and depth 10ft. If the tank is full how much work will it take to pump out the water.

3) 8.5.12 A water tank which is in the shape of a right circular cylinder with height 20 ft and radius 6ft.

If the tank is half full of water how much work will it take to pump the water to the the rim of the tank?
How much work will it take if the tank is full of water?

4) 8.5.18 A cylindrical barrel, standing upright, contains muddy water. The top of the barrel, which has diameter 1 meter, is open. The height of the barrel is 2m and it is filled to a depth of 1m. The density of the water below the surface is given by rho(h) = 1 = kh kg/m^3 where k is a positive constant. Find the total work to pump all of the water to the top rim of the barrel.

5) 8.5.20 A reservoir has a dam on one end. The dam is a rectangular wall which is 1500ft long and is 75ft high.

Approximate the total force on the wall with a Riemann sum.
Write an integral which represents the force and evaluate it.

6) A glass is in the shape of an inverted right circular cone. It has height 10cm and top width 10cm. How much work does it take to raise the water in the glass to 15cm above the apex of the cone if the glass is full.

7) 8.5.30 Find the kinetic energy of a phonograph record of uniform density, mass 50g, and radius 10cm rotating at 33+1/3 revolutions per minute. Note that kinetic energy is KE= (1/2)mv^2 for a particle of mass m and at velocity v. (Hint: try to slice the object in such a way that KE is uniform on the slice.)

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Section 8.6

1) 8.6.6 A person won the lottery and was given the choice between $200 million paid out over 25yrs or a lump sum of $105 million paid immediately.

Which option is better if interest is compounded continuously at 6%? What about 5%
The winner chose the lump sum option. What assumption was she making about interest rates?

2) 8.6.8 How long does it take an investment of $1000/yr compounded continuously at 5% to become $10000. How long if the initial investment is $2000?

3) 8.6.12 The value of (good) wine increases with age. Therefore, if you are a wine dealer, you have the problem of deciding whether to sell your wine now, at the price of $P per bottle, or to sell it later at a higher price. Suppose you know that a wine-drinker is willing to pay for a bottle of this wine t years from now is $P(1 + 15sqrt(t)). Assuming continuous compounding and an interest rate of 5% per year. When is the best time to sell your wine?

4) 8.6.16 Using Riemann sums, explain the economic significance of Int((p*-S(q)) dq ,0,q*) to the producers.

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Section 8.7

1) 8.7.2 Graph and describe a density function and a cumulative distribution function which could represent the distribution of income through a population which has small middle and upper classes and a large amount of poor people.

2) 8.7.4 y = 4 when x is in [0,c]. y = 0 otherwise.

Is this function a PDF or CDF? Why?
Find the appropriate value of c.
If the function was a PDF, find the corresponding CDF. If the function was a CDF, find the corresponding PDF.

3) 8.7.6 y = (5/c)x on [0,c] and y = c on [c,inf]

Is this function a PDF or CDF? Why?
Find the appropriate value of c.
If the function was a PDF, find the corresponding CDF. If the function was a CDF, find the corresponding PDF.

4) 8.7.8 The function y consists of the line segments (0,0) to (2,c) to (4,3c) to (?, 3c)

Is this function a PDF or CDF? Why?
Find the appropriate value of c.
If the function was a PDF, find the corresponding CDF. If the function was a CDF, find the corresponding PDF.

5) 8.7.12 Suppose that the CDF for heights (in meters) of trees in a forest is F(x)

Explain in terms of trees the meaning of the statement F(7) = 0.6.
Which is greater, F(7) or F(6)? Why?

6) What is the PDF for the position of a pendulum while swinging?

7) 8.7.20 Consider a population of individuals with some disease. Suppose that t is the number of years since the onset of the disease and the death density function is given by f(t) = cte^(-kt). The fraction who die during a time interval [t, t+`dt] can be approximated by f(t)`dt = cte^(-kt)`dt. c and k are positive constants depending on the certain disease.

Find the value of c in terms of k.
If 40% of the population dies within 5 years, find c and k.
Find the CDF of the distribution and give the function in terms of k.

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