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mathematics
calculus

Calculus 9th edition Dale Varberg, Edwin J. Purcell, Steven E. Rigdon - Solutions

Prove the formula ∫ fm-1 (x)gn-1(x) [nf(x) g'(x) + mg(x) f'(x)] dx = fm(x) gn(x) + C
Evaluate the indefinite integral ∫ sin3[(x2 + 1)] cos [(x2 + 1)4] (x2 + 1)3 x dx
Some software packages can evaluate indefinite integrals. Use your software on each of the following. a. ∫ 6 sin(3(x - 2)) dx b. ∫ sin3(x/6) dx c. ∫ (x2 cox 2x + x sin 2x) dx
Let F0(x) = x sin x and Fn+1(x) = ∫ Fn(x) dx. a. Determine F1(x), F2(x), F3(x), and F4(x). b. On the basis of part (a), conjecture the form of F16(x).
In Problems a-c, show that the indicated function is a solution of the given differential equation; that is substitute the indicated function for y to see that it produces an equality. a. dy/dx + x/y = 0; y = √1 - x2 b. - x dy/dx + y = 0; y = Cx c. d2y/dx2 + y = 0; y = C1 sin x + C2 cos x
Find the xy-equation of the curve through (1, 2) whose slope at any point is three times its x-coordinate
Find the xy-equation of the curve through (1, 2) whose slope at any point is three times the square of its y-coordinate.
In Problems a to b, an object is moving along a coordinate line subject to the indicated acceleration a (in centimeters per second per second) with the initial velocity vo (in centimeters per second) and directed distance so (in centimeters). Find both the velocity and directed distance s after 2
A ball is thrown upward from the surface of the earth with an initial velocity of 96 feet per second. What is the maxi-mum height that it reaches?
A ball is thrown upward from the surface of a planet where the acceleration of gravity is k (a negative constant) feet per second per second. If the initial velocity is vo, show that the maximum height is -v20/2k.
On the surface of the moon, the acceleration of gravity is -5.28 feet per second per second. If an object is thrown upward from an initial height of 1000 feet with a velocity of 56 feet per second, find its velocity and height 4.5 seconds later.
What is the maximum height that the object of Problem 23 reaches? On the surface of the moon, the acceleration of gravity is -5.28 feet per second per second. If an object is thrown upward from an initial height of 1000 feet with a velocity of 56 feet per second, find its velocity and height 4.5
The rate of change of volume V of a melting snowball is proportional to the surface area S of the ball; that is, dV/dt = -kS, where k is a positive constant. If the radius of the ball at t = 0 is r = 2 and at t = 10 is r = 0.5, show that r = 3/20 t + 2.
From what height above the earth must a ball be dropped in order to strike the ground with a velocity of -136 feet per second?
Determine the escape velocity for an object launched from each of the following celestial bodies. Here g ‰ˆ 32 feet per second per second.
If the brakes of a car, when fully applied, produce a constant deceleration of 11 feet per second per second, what is the shortest distance in which the car can be braked to a halt from a speed of 60 miles per hour?
A certain rocket, initially at rest, is shot straight up with an acceleration of 6t meters per second per second during the first 10 seconds after blast-off, after which the engine cuts out and the rocket is subject only to gravitational acceleration of -10 meters per second per second. How high
Starting at station A, a commuter train accelerates at 3 meters per second per second for 8 seconds, then travels at constant speed vm for 100 seconds, and finally brakes (decelerates) to a stop at station B at 4 meters per second per second. Find a. vm and b. The distance between A and B.
Starting from rest, a bus increases speed at constant acceleration a1, then travels at constant speed vm and finally brakes to a stop at constant acceleration a2 (a2 < 0). It took 4 minutes to travel the 2 miles between stop C and stop D and then 3 minutes to go the 1.4 miles between stop D and
A hot-air balloon left the ground rising at 4 feet per second. Sixteen seconds later, Victoria threw a ball straight up to her friend Colleen in the balloon. At what speed did she throw the ball if it just made it to Colleen?
According to Torricelli's Law, the time rate of change of the volume V of water in a draining tank is proportional to the square root of the water's depth. A cylindrical tank of radius 10/√π centimeters and height 16 centimeters, which was full initially, took 40 seconds to drain. a. Write the
The wolf population Pin a certain state has been growing at a rate proportional to the cube root of the population size. The population was estimated at 1000 in 1980 and at 1700 in 1990. a. Write the differential equation for P at time t with the two corresponding conditions. b. Solve the
At t = 0, a ball was dropped from a height of 16 feet. It hit the floor and rebounded to a height of 9 feet Figure 4.a. Find a two-part formula for the velocity v(t) that is valid until the ball hits the floor a second time. b. At what two times was the ball at height 9 feet?
In Problems a-b, first find the general solution (involving a constant C) for the given differential equation. Then find the particular solution that satisfies the indicated condition. a. dy/dx = x2 + 1; y = 1 at x = 1 b. dy/dx = x-3 + 2; y = 3 at x = 1
In Problems a-e, find the area of the shaded region.a.b. c. d. e.
In Problems a-b, a function f and its domain are given. Deter-mine the critical points, evaluate f at these points, and find the (global) maximum and minimum values. a. (x) = x2 - 2x; [0, 4] b. f(t) =1/t;[1, 4]
In Problems a to c, a function f is given with domain (-∞, ∞), Indicate where f is uncreasing and where it is concave down. a. f(x) = 3x - x2 b. f(x) = x9 c. f(x) = x3 - 3x + 3
Find where the function g, defined by g(t) = t3 + l/t, is increasing and where it is decreasing. Find the local extreme values of g. find the point of inflection. Sketch the graph.
Find where the function f. defined by f(x) = x2(x - 4). is increasing and where it is decreasing. Find the local extreme values of f. Find the point of inflection. Sketch the graph.
Find the maximum and minimum values, if they exist, of the function defined by f(x) = 4/x2 + 1 + 2
In Problems a-b, sketch the graph of the given function f, labeling all extrema (local and global) and the inflection points and showing any asymptotes Be sure to make use of f' and f". a. f(x) = x4 - 2x b. f(x) = (x2 - 1)2
In Problems a to b, sketch the graph of the given function fin the region (-π, π), unless otherwise indicated, labeling all extrema (local and global) and the inflection points and showing any asymptotes. Be sure to make use of f' and f". a. f(x) = cos x - sin x b. f(x) = sin x - tan x
Sketch the graph of a function F that has all the following properties: a. F is everywhere continuous; b. F(-2) = 3, F(2) = -1; c. F'(x) = 0 for x > 2; d. F"(x) < 0 for x < 2
Sketch the graph of a function F that has all the following properties: a. F is everywhere continuous; b. F(-1) = 6, F(3) = -2; c. F'(x) < 0 for x < -1, F'(-1) = F'(3) = -2, F(7) = 0; d. F"(x) < 0 for x < -1, F"(x) = 0 for -1 < x < 3, F"(x) > 0 for x > 3.
Sketch the graph of a function F that has the following properties: a. F is everywhere continuous; b. F has period π; c. 0 ≤ F(x) ≤ 2, F(0) = 0, F(π/2) = 2; d. F'(x) > 0 for 0 < x < (π/2), F'(x) < 0 for π/2 < x < π; 2 (e) F"(x) < 0 for 0 < x < π.
A long sheet of meta1, 16 inches wide, is to be turned up at both sides to make a horizontal gutter with vertical sides. How many inches should be turned up at each side for maximum carrying capacity?
A fence, 8 feet high, is parallel to the wall of a building and 1 foot from the building. What is the shortest plank that can go over the fence, from the level ground, to prop the wall?
A page of a book is to contain 27 square inches of print. If the margins at the top, bottom, and one side are 2 inches and the margin at the other side is 1 inch, what size page would use the least paper?
A metal water trough with equal semicircular ends and open top is to have a capacity of 128Ï€ cubic feet Figure 1. Determine its radius r and length h if the trough is to require the least material for its construction.
Find the maximum and the minimum of the function defined on the closed interval [-2, 2] byFind where the graph is concave up and where it is concave down. Sketch the graph.
For each of the following functions, decide whether the Mean Value Theorem applies on the indicated interval I. If so, find all possible values of c, if not, tell why. Make a sketch. a. f(x) = x3/3; I =[-3, 3] b. F(x) = x3/5 + 1; I = [-1, 1] c. g(x) = x + 1/x - 1; I - [2, 3]
Find the equations of the tangent lines at the inflection points of the graph of y = x4 - 6x3 + 12x2 - 3x + 1
Let f be a continuous function with f(1) = -1/4, f(2) = 0, and f (3) = -1/4. If the graph of y f'(x) is as shown in Figure 2, sketch a possible graph for y = f(x).
Sketch the graph of a function G with all the following properties:a. G(x) is continuous and G"(x) > 0 for all x in (-ˆž, 0) ˆª (0, ˆž);b. G(-2) = G(2) = 3;c.d.
Use the Bisection Method to solve 3x - cos 2x = 0 ac curate to six decimal places. Use at a1 = 0 and b1 = 1.
Use Newton's Method to solve 3x - cos2x = 0 accurate to six decimal places. Use x1 = 0.5.
Use the Fixed-Point Algorithm to solve 3x - cos2x = 0, starting with .x1 = 0.5.
Use Newton's Method to find the solution of x - tan x = 0 in the interval (π, 2π.) accurate to four decimal places. Sketch graphs of y = x and y = tan x using the same axes to get a good initial guess for x1.
In Problems a-b, evaluate the indicated intergrals. a. ∫ (x3 - 3x2 + 3√x) dx b. ∫ 2x4 - 3x2 + 1 /x2 dx
In Problems a-c, solve the differential equation subject to the indicated condition. a. dy/dx = sin s; y = 2 at x = 0 b. dy/dx =1/√x + 1; y = 18 at x = 3 c. dy/dx = csc y; y = π at x = 0
A ball is thrown directly upward from a tower 448 feet high with an initial velocity of 48 feet per second. In how many seconds will it strike the ground and with what velocity? Assume that g = 32 feet per second per second and neglect air resistance.
In problems, find the value of the indicated sum.(a)(b) (c)
In problems suppose thatCalculate each of the following (a) (b) (c)
In problems use special sum formulas 1-4 to find each sum(a)(b) (c)
Add both sides of the two equalities below, solve for S and thereby give another proof of Formula 1. S = 1 + 2 + 3 + ... + (n-2) + (n-1) + n S = n+ (n-1) + (n-2) + ... + 3 + 2 + 1
Prove the following formula for a geometric sum
Use problem 26 to calculate each sum(a)(b) In problem 26 Prove the following formula for a geometric sum
Use a derivation like that in Problem 25 to obtain a formula for the arithmetic sum.In problem 25 S = 1 + 2 + 3 + ... + (n-2) + (n-1) + n S = n+ (n-1) + (n-2) + ... + 3 + 2 + 1
Use the identity (I + 1)3 - i3 = 3i2 + 3i + 1 to prove special sum formula 2
Use the identity (i+1)4 - i4 = 4i3 + 6i2 + 4i + 1 to prove special sum formula 3.
Use the identity (i+1)5 - i5 = 5i4 + 10i3 + 10i2 + 5i + 1 to prove special sum formula 1 and 3.
Use the diagrams in Figure 12 to establish formulas 1 and 3
In statistics we defined the mean x and the variance s2 of sequence of number x1, x2, ... xn byFind x and s2 for the sequence of numbers 2, 5, 7, 8, 9, 10, 14
Using the definition is problem 33, find x and s3 for each sequence of number.(a). 1,1,1,1,1(b) 1001, 1001, 1001, 1001, 1001(c). 1,2 ,3(d) 1000001, 1000002, 1000003In problem 33
Use the definition in Problem 33 to show that each in true.(a)(b) In problem 33
Based on your response to parts (a) and (b) of problem 34 make a conjecture about the variance of n identical numbers prove your conjecture.
Let x1, x2,....xn be any real numbers. Find the value of c that minimizes
In the song The Twelve Days of Christmas, my true love gave me 1 gift on the first day, 1 + 2 gifts on the second day, 1 + 2 + 3 gifts on the third day, and so on for 12 days. (a) Find the total number of gifts given in 12 days. (b) Find a simple formula for Tn, the total number of gifts given
A grocer stacks oranges in a pyramid like pile. If the bottom layer is rectangular with 10 rows of 16 oranges and the top layer has a single row of oranges, how many oranges are in the stack?
Generalize the result of Problems 39 and 40 to the case of m rows of n oranges.
Find a nice formula for sum
In problems find the area of the indicated inscribed or circumscribed polygon.(a)(b) (c) (d)
In Problems, sketch the graph of the given function over the interval [a, b]; then divide [a, b] into n equal subintervals Finally, calculate the area of the corresponding circumscribed polygon. (a). f(x) = x + 1; a = -1, b = 2, n = 3 (b). f(x) = 3x - 1; a = 1, b = 3, n = 4 (c). f (x) = x2 - 1; a
In Problems, find the area of the region under the curve y = f (x) over the interval [a, b]. To do this, divide the interval [a, b) into n equal subintervals, calculate the area of the corresponding circumscribed polygon, and then let n →∞. (See the example for y = x2 in the text) (a) y = x +
Suppose that an object is traveling along the x-axis in such a way that its velocity at time t seconds is v = t + 2 feet per second. How far did it travel between t = 0 and t = 1?
Follow the directions of Problem 59 given that v = 1/2t2 + 2. You may use the result of Problem 54.
Let Aba, denote the area under the curve y = x2 over the interval [a, b]. (a) Prove that Ab0 = b3/3 (b) Show that Aba = b3/3 - a3/3. Assume that a ≥ 0.
Use the results of Problem 61 to calculate the area under the curve y = x2 over each of the following intervals. (a) [0, 5] (b) [1, 4] (c) [2, 5]
Use the result of Problem 64 to calculate each of the following areas. (a). A20 (x3) (b). A21(x3) (c). A21(x5) (d). A20(x9)
In problems write the indicated sum in sigma notation. (a) 1 + 2 + 3 + .....+ 41 (b) 2+4+6+8+....+ 50 (c) 1 + ½ + 1/3 + ¼ + .... 1/100
In problems calculate the Riemann sum suggested by each figure.(a)(b)
In problems evaluate the definite integrals using the definition, as in Example3 and 4.(a)(b)
In problems calculateWhere a and b are the left and right end points for which f is defined, by using the interval Additive property and the appropriate area formulas from plane geometry. Begin by graphing the given function (a) (b) (c)
In problems the velocity for an object is given Assuming that the object is at the origin at time t = 0, find the position at time t = 4(a) v(t) = t/60(b) v(t) = 1 + 3t(c)(d)
In problems, an object's velocity function is graphed. Use this graph to determine the object's position at times t = 20, 40, 60, 80, 100 and 120 assuming the object is at the origin at time t = 0(a)(b) (c) (d)
In problems, calculate the Riemann sumFor the given data (a) f(x) = x - 1; P: 3 x1 = 3, x2 = 4, x3 = 4.75, x4 = 6, x5 = 6.5 (b). f(x) = -x/2 + 3; P: -3 x1 = -2, x2 = -0.5 x3 = x4 = 2 (c). f(x) = x2/2 + x; [-2, 2J is divided into eight equal subintervals. I; is the midpoint.
Recall that [x] denotes the greatest integer less than or equal to x. Calculate each of the following integrals. You many use geometric reasoning and the fact that(a) (b) (c) (d) (e) (f) (g) (h)
Let f be an odd function and g be an even function and suppose thatUse geometric reasoning to calculate each of the following (a) (b) (c) (d) (e) (f)
Show thatBy completing the following argument for the partition a = x0 Now simplify RP.
Show thatBy an argument like that in problem, but using xi = [1/2(xi-1 + xi-1xi + x2i[1/2 assume that 0 ‰¤ a
Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system in Problems 35-38, evaluate the 10-subinterval Riemann sums using left end point, right end point, and midpoint evaluations(a)(b) (c)
Prove that the function f defined byIs not integrable on [0, 1]
In problems use the given values of a and b and express the given limit as a definite integral.(a)a = 1, b = 3 (b) a = 0, b = 2 (c) a = -1, b = 1 (d) a = 0, b = Ï€
In problems, find a formula for and graph the accumulation function A(x) that is equal to the indicated area.(a)(b) (c)
In problems, find the interval(s) on which the graph of y = f(x), x ‰¥ 0, is (a) increasing and (b) concave up.(a)(b) (c)
In problems, use the interval Additive property and linearity to evaluateBegin by drawing a graph of f. (a) (b) (c) f(x) = 3 + |x - 3|
Consider the function G(x) = ˆ«x0 f (t) dt, where f(t) oscillates about the line y = 2 over the r-region [0, 101 and is given by Figure 10.(a) At what values of x over this region do the local maxima and minima of G(x) occur?(b) Where does G(x) attain its absolute maximum and absolute
Perform the same analysis as you did in Problem 37 for the function G(x) = ˆ«x0 f (t) dt given by Figure 11, where f(t) oscillates about the line y = 2 for the interval [0,10].In problem 37 (a) At what values of x over this region do the local maxima and minima of G(x) occur? (b) Where
In problems use a graphing calculator to graph each integrand. Then use the Boundedness property find a lower bound and an upper bound for each definite integral.(a)(b) (c)
In problems decide whether the given statement is true or false. Then justify your answer.(a) If f is continuous and f(x) ‰¥ 0 for all x in [a, b] then(b) If Then f(x) ‰¥ 0 for all x in 'a, b] (c) If Then f(x) = 0 for all x in [a, b]

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