In Exercises 110, evaluate the integral analytically. Then

Chapter 6: Problem 1 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 110, evaluate the integral analytically. Then use NINT to support your result.

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Chapter 6: Problem 2 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 110, evaluate the integral analytically. Then use NINT to support your result.

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Chapter 6: Problem 3 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 110, evaluate the integral analytically. Then use NINT to support your result.

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Chapter 6: Problem 4 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 110, evaluate the integral analytically. Then use NINT to support your result.

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Chapter 6: Problem 5 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 110, evaluate the integral analytically. Then use NINT to support your result.

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Chapter 6: Problem 6 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 110, evaluate the integral analytically. Then use NINT to support your result.

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Chapter 6: Problem 7 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 110, evaluate the integral analytically. Then use NINT to support your result.

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Chapter 6: Problem 8 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 110, evaluate the integral analytically. Then use NINT to support your result.

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Chapter 6: Problem 9 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 110, evaluate the integral analytically. Then use NINT to support your result.

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Chapter 6: Problem 10 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 110, evaluate the integral analytically. Then use NINT to support your result.

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Chapter 6: Problem 11 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 12 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 13 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 14 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 15 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 16 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 17 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 18 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 19 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 20 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 21 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 22 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 23 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 24 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 1124, evaluate the integral.

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Chapter 6: Problem 25 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 2534, solve the initial value problem analytically. Support your solution by overlaying its graph on a slope field of the differential equation.

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Chapter 6: Problem 26 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 2534, solve the initial value problem analytically. Support your solution by overlaying its graph on a slope field of the differential equation.

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Chapter 6: Problem 27 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 2534, solve the initial value problem analytically. Support your solution by overlaying its graph on a slope field of the differential equation.

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Chapter 6: Problem 28 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 2534, solve the initial value problem analytically. Support your solution by overlaying its graph on a slope field of the differential equation.

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Chapter 6: Problem 29 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 2534, solve the initial value problem analytically. Support your solution by overlaying its graph on a slope field of the differential equation.

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Chapter 6: Problem 30 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 2534, solve the initial value problem analytically. Support your solution by overlaying its graph on a slope field of the differential equation.

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Chapter 6: Problem 31 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 2534, solve the initial value problem analytically. Support your solution by overlaying its graph on a slope field of the differential equation.

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Chapter 6: Problem 32 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 2534, solve the initial value problem analytically. Support your solution by overlaying its graph on a slope field of the differential equation.

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Chapter 6: Problem 33 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 2534, solve the initial value problem analytically. Support your solution by overlaying its graph on a slope field of the differential equation.

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Chapter 6: Problem 34 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 2534, solve the initial value problem analytically. Support your solution by overlaying its graph on a slope field of the differential equation.

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Chapter 6: Problem 35 Calculus: Graphical, Numerical, Algebraic 3

Find an integral equation y % ! x a f (t)dt such that dy/dx % sin3 x and y % 5 when x % 4.

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Chapter 6: Problem 36 Calculus: Graphical, Numerical, Algebraic 3

Find an integral equation y % ! x a f (t)dt such that dy/dx % "1#" x4 and y % 2 when x % 1.

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Chapter 6: Problem 37 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 37 and 38, construct a slope field for the differential equation. In each case, copy the graph shown and draw tiny segments through the twelve lattice points shown in the graph. Use slope analysis, not your graphing calculator.

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Chapter 6: Problem 38 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 37 and 38, construct a slope field for the differential equation. In each case, copy the graph shown and draw tiny segments through the twelve lattice points shown in the graph. Use slope analysis, not your graphing calculator.

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Chapter 6: Problem 39 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3942, match the differential equation with the appropriate slope field. (All slope fields are shown in the window [$6, 6] by [$4, 4].)

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Chapter 6: Problem 40 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3942, match the differential equation with the appropriate slope field. (All slope fields are shown in the window [$6, 6] by [$4, 4].)

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Chapter 6: Problem 41 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3942, match the differential equation with the appropriate slope field. (All slope fields are shown in the window [$6, 6] by [$4, 4].)

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Chapter 6: Problem 42 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 3942, match the differential equation with the appropriate slope field. (All slope fields are shown in the window [$6, 6] by [$4, 4].)

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Chapter 6: Problem 43 Calculus: Graphical, Numerical, Algebraic 3

Suppose dy/dx ! x " y # 1 and y ! 1 when x ! 1. Use Eulers Method with increments of x ! 0.1 to approximate the value of y when x ! 1.3

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Chapter 6: Problem 44 Calculus: Graphical, Numerical, Algebraic 3

Suppose dy/dx ! x # y and y ! 2 when x ! 1. Use Eulers Method with increments of x ! #0.1 to approximate the value of y when x ! 0.7

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Chapter 6: Problem 45 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 45 and 46, match the indefinite integral with the graph of one of the antiderivatives of the integrand

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Chapter 6: Problem 46 Calculus: Graphical, Numerical, Algebraic 3

In Exercises 45 and 46, match the indefinite integral with the graph of one of the antiderivatives of the integrand

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Chapter 6: Problem 47 Calculus: Graphical, Numerical, Algebraic 3

The figure shows the graph of the function y ! f "x# that is the solution of one of the following initial value problems. Which one? How do you know?

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Chapter 6: Problem 48 Calculus: Graphical, Numerical, Algebraic 3

Does the following initial value problem have a solution? Explain

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Chapter 6: Problem 49 Calculus: Graphical, Numerical, Algebraic 3

The acceleration of a particle moving along a coordinate line is $ d dt 2 s $2 ! 2 " 6t m%sec2. At t ! 0 the velocity is 4 m%sec. (a) Find the velocity as a function of time t. v ! 2t " 3t 2 " 4 (b) How far does the particle move during the first second of its trip, from t ! 0 to t ! 1?

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Chapter 6: Problem 50 Calculus: Graphical, Numerical, Algebraic 3

Draw a possible graph for the function y ! f "x# with slope field given in the figure that satisfies the initial condition y"0# ! 0.

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Chapter 6: Problem 51 Calculus: Graphical, Numerical, Algebraic 3

What costs $27 million per gram and can be used to treat brain cancer, analyze coal for its sulfur content, and detect explosives in luggage? The answer is californium-252, a radioactive isotope so rare that only about 8 g of it have been made in the western world since its discovery by Glenn Seaborg in 1950. The half-life of the isotope is 2.645 yearslong enough for a useful service life and short enough to have a high radioactivity per unit mass. One microgram of the isotope releases 170 million neutrons per second. (a) What is the value of k in the decay equation for this isotope? (b) What is the isotopes mean life? (See Exercise 19, Section 6.4.)

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Chapter 6: Problem 52 Calculus: Graphical, Numerical, Algebraic 3

A deep-dish apple pie, whose internal temperature was 220F when removed from the oven, was set out on a 40F breezy porch to cool. Fifteen minutes later, the pies internal temperature was 180F. How long did it take the pie to cool from there to 70F?

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Chapter 6: Problem 53 Calculus: Graphical, Numerical, Algebraic 3

A pan of warm water "46C# was put into a refrigerator. Ten minutes later, the waters temperature was 39C; 10 minutes after that, it was 33C. Use Newtons Law of Cooling to estimate how cold the refrigerator was

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Chapter 6: Problem 54 Calculus: Graphical, Numerical, Algebraic 3

painting attributed to Vermeer "16321675#, which should contain no more than 96.2% of its original carbon-14, contains 99.5% instead. About how old is the forgery?

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Chapter 6: Problem 55 Calculus: Graphical, Numerical, Algebraic 3

What is the age of a sample of charcoal in which 90% of the carbon-14 that was originally present has decayed?

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Chapter 6: Problem 56 Calculus: Graphical, Numerical, Algebraic 3

A violin made in 1785 by John Betts, one of Englands finest violin makers, cost $250 in 1924 and sold for $7500 in 1988. Assuming a constant relative rate of appreciation, what was that rate?

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Chapter 6: Problem 57 Calculus: Graphical, Numerical, Algebraic 3

The intensity L!x" of light x feet beneath the surface of the ocean satisfies the differential equation ! d d L x ! " #kL, where k is a constant. As a diver you know from experience that diving to 18 ft in the Caribbean Sea cuts the intensity in half. You cannot work without artificial light when the intensity falls below a tenth of the surface value. About how deep can you expect to work without artificial light?

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Chapter 6: Problem 58 Calculus: Graphical, Numerical, Algebraic 3

Under certain conditions, the result of the movement of a dissolved substance across a cells membrane is described by the equation ! d d y t ! " k ! A V ! !c # y", where y is the concentration of the substance inside the cell, and dy$dt is the rate with which y changes over time. The letters k, A, V, and c stand for constants, k being the permeability coefficient (a property of the membrane), A the surface area of the membrane, V the cells volume, and c the concentration of the substance outside the cell. The equation says that the rate at which the concentration changes within the cell is proportional to the difference between it and the outside concentration. (a) Solve the equation for y!t", using y0 " y!0". (b) Find the steady-state concentration, limt% y!t".

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Chapter 6: Problem 59 Calculus: Graphical, Numerical, Algebraic 3

The spread of flu in a certain school is given by the formula P!t" "!1 & 15 e 0 !4.3#t , where t is the number of days after students are first exposed to infected students. (a) Show that the function is a solution of a logistic differential equation. Identify k and the carrying capacity. k " 1; carrying (b) Writing to Learn Estimate P!0". Explain its meaning in the context of the problem. (c) Estimate the number of days it will take for a total of 125 students to become infected. A

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Chapter 6: Problem 60 Calculus: Graphical, Numerical, Algebraic 3

Show that y " # x 0 sin !t 2" dt & x3 & x & 2 is the solution of the initial value problem. Differential equation: y' " 2x cos !x2" & 6x Initial conditions: y(!0" " 1, y!0" " 2

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Chapter 6: Problem 61 Calculus: Graphical, Numerical, Algebraic 3

Use analytic methods to find the exact solution to ! d d P t ! " 0.002P(1 # !8 P 00!), P!0" " 50.

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Chapter 6: Problem 62 Calculus: Graphical, Numerical, Algebraic 3

Give two ways to provide graphical support for the integral formula #x2 ln x dx " ! x 3 3 ! ln x # ! x 9 3 ! & C.

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Chapter 6: Problem 63 Calculus: Graphical, Numerical, Algebraic 3

Find the amount of time required for $10,000 to double if the 6.3% annual interest is compounded (a) annually, (b) continuously.

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Chapter 6: Problem 64 Calculus: Graphical, Numerical, Algebraic 3

Let f !x" " # x 0 u!t" dt and g!x" " # x 3 u!t" dt. (a) Show that f and g are antiderivatives of u!x". (b) Find a constant C so that f !x" " g!x" & C.

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Chapter 6: Problem 65 Calculus: Graphical, Numerical, Algebraic 3

Table 6.9 shows the population of Anchorage, AK for selected years between 1950 and 2003. (a) Use logistic regression to find a curve to model the data and superimpose it on a scatter plot of population against years after 1950. (b) Based on the regression equation, what number will the Anchorage population approach in the long run? (c) Write a logistic differential equation in the form dp/dt " kP(M # P) that models the growth of the Anchorage data in Table 6.9.

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Chapter 6: Problem 66 Calculus: Graphical, Numerical, Algebraic 3

A temperature probe is removed from a cup of hot chocolate and placed in water whose temperature !Ts" is 0C. The data in Table 6.10 were collected over the next 30 sec with a CBL temperature probe. (a) Find an exponential regression equation for the !t, T " data. Superimpose its graph on a scatter plot of the data. (b) Estimate when the temperature probe will read 40C. (c) Estimate the hot chocolates temperature when the temperature probe was removed.

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Chapter 6: Problem 67 Calculus: Graphical, Numerical, Algebraic 3

The spread of a rumor through a small town is modeled by dy/dt ! 1.2y(1 " y), where y is the proportion of the townspeople who have heard the rumor at time t in days. At time t ! 0, ten percent of the townspeople have heard the rumor. (a) What proportion of the townspeople have heard the rumor when it is spreading the fastest? 1/2 (b) Find y explicitly as a function of t. (c) At what time t is the rumor spreading the fastest?

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Chapter 6: Problem 68 Calculus: Graphical, Numerical, Algebraic 3

A population P of wolves at time t years (t # 0) is increasing at a rate directly proportional to 600 " P, where the constant of proportionality is k. (a) If P(0) ! 200, find P(t) in terms of t and k. (b) If P(2) ! 500, find k. (c) Find limt$ P(t).

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Chapter 6: Problem 69 Calculus: Graphical, Numerical, Algebraic 3

Let v(t) be the velocity, in feet per second, of a skydiver at time t seconds, t # 0. After her parachute opens, her velocity satisfies the differential equation dv/dt ! "2(v % 17), with initial condition v(0) ! "47. (a) Use separation of variables to find an expression for v in terms of t, where t is measured in seconds. (b) Terminal velocity is defined as limt$ v(t). Find the terminal velocity of the skydiver to the nearest foot per second. (c) It is safe to land when her speed is 20 feet per second. At what time t does she reach this speed?

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