Math

CalculusFinite Mathematics and Applied Calculus (MindTap Course List)End of the Earth In 5 billion year the Sun will have run out of hydrogen fuel and begin to expand into a red giant, eventually engulfing the Earth and causing it to spiral into the core of the Sun 7.5 billion years from now. The following graph 50 shows the expanding radius of the red giant Sun (in red) and the radius of the Earth’s orbit about the Sun (in green) during its final three and a half million years of existence. The radii are measured in astronomical units (AU; one AU is the current radius of the Earth’s orbit around the Sun, approximately 93 million miles), and time is measured in millions of years. r (AU) The curve representing the Sun’s radius has equation r = 0.037 t 2 + 0.02 t + 0.4. ( t = 4 marks the end of the red giant expansion phase.) Exercises 45and 46are based on this curve. a. Calculate the rate of change of the radius of the Sun over the successive intervals [ 0 , 2 ] , [ 1 , 3 ] , [ 2 , 4 ] . b. The successive rates of change are a linear function of t . What is the slope of that linear function? How fast will the rate of change of the Sun’s radius be increasing in the final 4 million years?BuyFind*arrow_forward*

7th Edition

Stefan Waner + 1 other

Publisher: Cengage Learning

ISBN: 9781337274203

Chapter 10.4, Problem 46E

Textbook Problem

**End of the Earth** In 5 billion year the Sun will have run out of hydrogen fuel and begin to expand into a red giant, eventually engulfing the Earth and causing it to spiral into the core of the Sun 7.5 billion years from now. The following graph^{50} shows the expanding radius of the red giant Sun (in red) and the radius of the Earth’s orbit about the Sun (in green) during its final three and a half million years of existence. The radii are measured in astronomical units (AU; one AU is the current radius of the Earth’s orbit around the Sun, approximately 93 million miles), and time is measured in millions of years.

*r* (AU)

The curve representing the Sun’s radius has equation

(

**a.** Calculate the rate of change of the radius of the Sun over the successive intervals

**b.** The successive rates of change are a linear function of *t*. What is the slope of that linear function? How fast will the *rate of change* of the Sun’s radius be increasing in the final 4 million years?

Expert Solution

**(a)**

To determine

**To calculate:** The rate of change of the radius of sun over the successive interval

**Given Information:**

The graph shows the expanding radius of the red giant Sun whose radii have equation with respect to time is

Where the radii are measured in astronomical units and time is measured in millions of years.

**Formula used:**

Average rate of change of

**Calculation:**

Considered the equation

Now, apply the average rate of change formula for the interval

Compute the value of

Now, substitute the value

Now, substitute the value

Substitute the value of

Now, apply the average rate of change formula for the interval

Compute the value of

Substitute the value

Expert Solution

**(b)**

To determine

**To calculate:** The slope of linear function

Finite Mathematics and Applied Calculus (MindTap Course List)

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In Exercises 4954, (a) use any method to estimate...Ch. 10.5 - In Exercises 55-58, estimate the given quantity....Ch. 10.5 - In Exercises 55-58, estimate the given quantity....Ch. 10.5 - In Exercises 55-58, estimate the given quantity....Ch. 10.5 - In Exercises 55-58, estimate the given quantity....Ch. 10.5 - In Exercises 5964, match the graph of f to the...Ch. 10.5 - In Exercises 5964, match the graph of f to the...Ch. 10.5 - In Exercises 5964, match the graph of f to the...Ch. 10.5 - In Exercises 5964, match the graph of f to the...Ch. 10.5 - In Exercises 5964, match the graph of f to the...Ch. 10.5 - In Exercises 5964, match the graph of f to the...Ch. 10.5 - In Exercises 6568 the graph of a function is...Ch. 10.5 - In Exercises 6568 the graph of a function is...Ch. 10.5 - In Exercises 6568 the graph of a function is...Ch. 10.5 - In Exercises 6568 the graph of a function is...Ch. 10.5 - In Exercises 6972 the graph of the derivative of a...Ch. 10.5 - In Exercises 6972 the graph of the derivative of a...Ch. 10.5 - In Exercises 6972 the graph of the derivative of a...Ch. 10.5 - In Exercises 6972 the graph of the derivative of a...Ch. 10.5 - In Exercises 73 and 74, use technology to graph...Ch. 10.5 - In Exercises 73 and 74, use technology to graph...Ch. 10.5 - In Exercises 75 and 76, use the method of Example...Ch. 10.5 - In Exercises 75 and 76, use the method of Example...Ch. 10.5 - Temperatures on Mars The air temperature one...Ch. 10.5 - Temperatures on Venus The air temperature one...Ch. 10.5 - Demand Suppose the demand for a new brand of...Ch. 10.5 - Demand Suppose the demand for an old brand of TV...Ch. 10.5 - Oil Imports from Mexico The following graph shows...Ch. 10.5 - Oil Production in Mexico The following graph shows...Ch. 10.5 - Prison Population The following curve is a model...Ch. 10.5 - Demand for Freon 12 The demand for...Ch. 10.5 - Velocity If a stone is dropped from a height of...Ch. 10.5 - Velocity If a stone is thrown down at 120 ft/s...Ch. 10.5 - Crude Oil Prices The price per barrel of crude oil...Ch. 10.5 - Median Home Prices The median home price in the...Ch. 10.5 - The 2003 SARS Outbreak In the early stages of the...Ch. 10.5 - The 2003 SARS Outbreak A few weeks into the SARS...Ch. 10.5 - The 2014 Ebola Outbreak In the first 6 months of...Ch. 10.5 - The 2014 Ebola Outbreak In the first 6 months of...Ch. 10.5 - Early Internet Services On January 1, 1996,...Ch. 10.5 - Early Internet Services On January 1, 1996,...Ch. 10.5 - Learning to Speak Let p(t) represent the...Ch. 10.5 - Learning to Read Let p(t) represent the percentage...Ch. 10.5 - Subprime Mortgages during the Housing Crisis (Com-...Ch. 10.5 - Subprime Mortgages during the Housing Crisis (Com-...Ch. 10.5 - Embryo Development The oxygen consumption of a...Ch. 10.5 - Embryo Development The oxygen consumption of a...Ch. 10.5 - The next two exercises are applications of...Ch. 10.5 - The next two exercises are applications of...Ch. 10.5 - In which, if any, of the following cases is f...Ch. 10.5 - In which, if any, of the following cases is f...Ch. 10.5 - Explain why we cannot put h=0 in the approximation...Ch. 10.5 - The balanced difference quotient...Ch. 10.5 - Let H(t) represent the number of Handbook members...Ch. 10.5 - Let F(t) represent the net earnings of Footbook,...Ch. 10.5 - It is now 8 months since the Garden City lacrosse...Ch. 10.5 - Having been soundly defeated in the national...Ch. 10.5 - Company As profits are given by P(0)=1 millionand...Ch. 10.5 - Company Cs profits are given by P(0)=1 millionand...Ch. 10.5 - During the 1-month period starting last January 1,...Ch. 10.5 - During the 1-month period starting last January 1,...Ch. 10.5 - If the derivative of f is zero at a point, what do...Ch. 10.5 - Sketch the graph of a function whose derivative...Ch. 10.5 - Sketch the graph of a function whose derivative...Ch. 10.5 - Sketch the graph of a function whose derivative is...Ch. 10.5 - Use the difference quotient to explain the fact...Ch. 10.5 - Give a numerical explanation of the fact that if f...Ch. 10.5 - Consider the following values of the function f...Ch. 10.5 - Consider the following values of the function g...Ch. 10.5 - Sketch the graph of a function whose derivative is...Ch. 10.5 - Sketch the graph of a function whose derivative is...Ch. 10.5 - Here is the graph of the derivative f ofa function...Ch. 10.5 - Here is the graph of the derivative f ofa function...Ch. 10.5 - Professor Talker of the physics department drove a...Ch. 10.5 - Professor Silent, another physics professor, drove...Ch. 10.5 - Draw the graph of a function f with the property...Ch. 10.5 - Draw the graph of a function f with the property...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 1-14, compute f(a) algebraically for...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 15-28, compute the derivative...Ch. 10.6 - In Exercises 29-38, compute the indicated...Ch. 10.6 - In Exercises 29-38, compute the indicated...Ch. 10.6 - In Exercises 29-38, compute the indicated...Ch. 10.6 - In Exercises 29-38, compute the indicated...Ch. 10.6 - In Exercises 29-38, compute the indicated...Ch. 10.6 - In Exercises 29-38, compute the indicated...Ch. 10.6 - In Exercises 29-38, compute the indicated...Ch. 10.6 - In Exercises 29-38, compute the indicated...Ch. 10.6 - In Exercises 29-38, compute the indicated...Ch. 10.6 - In Exercises 29-38, compute the indicated...Ch. 10.6 - In Exercises39-44, find the equation of the...Ch. 10.6 - In Exercises39-44, find the equation of the...Ch. 10.6 - In Exercises39-44, find the equation of the...Ch. 10.6 - In Exercises39-44, find the equation of the...Ch. 10.6 - In Exercises39-44, find the equation of the...Ch. 10.6 - In Exercises39-44, find the equation of the...Ch. 10.6 - Velocity If a stone is dropped from a height of...Ch. 10.6 - Velocity If a stone is dropped from a height of...Ch. 10.6 - Oil Imports from Mexico Daily crude oil imports to...Ch. 10.6 - Oil Imports from Mexico Daily crude oil production...Ch. 10.6 - Bottled Water Sales The following chart shows the...Ch. 10.6 - Bottled Water Sales The following chart shows...Ch. 10.6 - Ecology Increasing numbers of manatees have been...Ch. 10.6 - SAT Scores by Income The following graph shown...Ch. 10.6 - Television Advertising The cost, in thousands of...Ch. 10.6 - Television Advertising (Compare Exercises 53.) The...Ch. 10.6 - Of the three methods (numerical, graphical, and...Ch. 10.6 - Explain why we cannot put h=0 in the formula...Ch. 10.6 - You just got your derivatives test back, and you...Ch. 10.6 - Your friend just got his derivatives test back and...Ch. 10.6 - Your other friend just got her derivatives test...Ch. 10.6 - Your third friend just got her derivatives test...Ch. 10.6 - Your friend Muffy claims that, because the...Ch. 10.6 - Use the balanced difference quotient formula,...Ch. 10.6 - A certain function f has the property that f(a)...Ch. 10.6 - One cannot put h=0 in the formula...Ch. 10 - In Exercise 1-4, numerically estimate whether the...Ch. 10 - In Exercise 1-4, numerically estimate whether the...Ch. 10 - In Exercise 1-4, numerically estimate whether the...Ch. 10 - In Exercise 1-4, numerically estimate whether the...Ch. 10 - In Exercises 5 and 6 the graph of a function f is...Ch. 10 - In Exercises 5 and 6 the graph of a function f is...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 730, calculate the limit...Ch. 10 - In Exercises 31-34, find the average rate of...Ch. 10 - In Exercises 31-34, find the average rate of...Ch. 10 - In Exercises 31-34, find the average rate of...Ch. 10 - In Exercises 31-34, find the average rate of...Ch. 10 - In Exercise 35-38 you given the graph of a...Ch. 10 - In Exercise 35-38 you given the graph of a...Ch. 10 - In Exercise 35-38 you given the graph of a...Ch. 10 - In Exercise 35-38 you given the graph of a...Ch. 10 - Let f have the graph shown. Select the correct...Ch. 10 - Let f have the graph shown. Select the correct...Ch. 10 - In Exercises 4144, use the definition of the...Ch. 10 - In Exercises 4144, use the definition of the...Ch. 10 - In Exercises 4144, use the definition of the...Ch. 10 - In Exercises 4144, use the definition of the...Ch. 10 - In Exercises 4548, use technology to graph the...Ch. 10 - In Exercises 4548, use technology to graph the...Ch. 10 - In Exercises 4548, use technology to graph the...Ch. 10 - In Exercises 4548, use technology to graph the...Ch. 10 - Stock Investments OHaganBooks.com CEO John OHagan...Ch. 10 - Stock Investments John OHagans golf partner Juan...Ch. 10 - Real Estate Marjory Duffin has persuaded John...Ch. 10 - Advertising Costs OHaganBooks.com has (on further...Ch. 10 - Sales Since the start of July, OHaganBooks.com has...Ch. 10 - Rising Sea Level Marjory Duffin recently purchased...Ch. 10 - Real Estate The following graph (see Exercise 51)...Ch. 10 - Advertising Costs The following graph (see...Ch. 10 - Sales OHaganBooks.com fits the curve...Ch. 10 - Sea Levels Marjory Duffin fit the curve...Ch. 10 - Excluding subsidies, what should the annual...Ch. 10 - Excluding subsidies, what should the annual...Ch. 10 - What is the marginal emission charge (derivative...Ch. 10 - We said that the revised policy provided an...Ch. 10 - What change in C(q) wouldmake 8 million tons the...Ch. 10 - If the scenario in Exercise 5 took place, what...Ch. 10 - Because of intense lobbying by the utility...Ch. 10 - Suppose that instead of imposing a fixed charge...

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