Calculus: Single And Multivariable
7th Edition
ISBN: 9781119444190
Author: Hughes-Hallett, Deborah; Mccallum, William G.; Gleason, Andrew M.; Connally, Eric; Kalaycioglu, Selin; Flath, Daniel E.; Lahme, Brigitte; Lomen, David O.; Lock, Patti Frazer; Lovelock, David; Morris, Jerry; Lozano, Guadalupe I.; Mumford, David; Quinney, D
Publisher: WILEY
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Textbook Question
Chapter 3.9, Problem 44E
In Problems 41–45, find a formula for the error E(x) in the tangent line approximation to the function near x = a. Using a table of values for E(x)∕(x − a) near x = a, find a value of k such that E(x)∕(x − a) ≈ k(x − a). Check that, approximately, k = f″(a)∕2 and that E(x) ≈ (f″(a)∕2)(x − a)2.
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For f(x) and g(x) given in Problems 35–38, find
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35. f(x) = 3x g(x) = x'
36. f(x) = Vx g(x) = 1/x
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38. f(x) = (x – 1)? g(x) = 1 – 2x
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Chapter 3 Solutions
Calculus: Single And Multivariable
Ch. 3.1 - Let f(x) = 7. Using the denition of the...Ch. 3.1 - Let f(x) = 17x+11. Use the denition of the...Ch. 3.1 - For Exercises 35, determine if the derivative...Ch. 3.1 - For Exercises 35, determine if the derivative...Ch. 3.1 - For Exercises 35, determine if the derivative...Ch. 3.1 - For Exercises 649, find the derivatives of the...Ch. 3.1 - For Exercises 649, find the derivatives of the...Ch. 3.1 - For Exercises 649, find the derivatives of the...Ch. 3.1 - For Exercises 649, find the derivatives of the...Ch. 3.1 - For Exercises 649, find the derivatives of the...
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Find f(x), f(x), and f(x). (b)...Ch. 3.1 - If p(t)=t3+2t2t+4, find d2p/dt2 and d3p/dt3.Ch. 3.1 - If w(x)=x+(1/x), find d2w/dx2 and d3w/dx3.Ch. 3.1 - For Problems 6267, determine if the derivative...Ch. 3.1 - For Problems 6267, determine if the derivative...Ch. 3.1 - For Problems 6267, determine if the derivative...Ch. 3.1 - For Problems 6267, determine if the derivative...Ch. 3.1 - For Problems 6267, determine if the derivative...Ch. 3.1 - For Problems 6267, determine if the derivative...Ch. 3.1 - If f(x)=(3x+8)(2x5), find f(x) and f(x).Ch. 3.1 - Find the equation of the line tangent to the graph...Ch. 3.1 - Find the equation of the line tangent to y=x2+3x5...Ch. 3.1 - Find the equation of the line tangent to f(x) at x...Ch. 3.1 - (a) Find the equation of the tangent line to f(x)...Ch. 3.1 - (a) Use Figure 3.8 to rank the quantities f (1), f...Ch. 3.1 - The graph of y=x39x216x+1 has a slope of 5 at two...Ch. 3.1 - On what intervals is the graph of f(x)=x44x3 both...Ch. 3.1 - For what values of x is the function y=x55x both...Ch. 3.1 - (a) Find the eighth derivative of...Ch. 3.1 - For the functions in Problems 7881: (a) Find the...Ch. 3.1 - For the functions in Problems 7881: (a) Find the...Ch. 3.1 - For the functions in Problems 7881: (a) Find the...Ch. 3.1 - For the functions in Problems 7881: (a) Find the...Ch. 3.1 - With t in years since 2016, the height of a sand...Ch. 3.1 - A rubber balloon contains neon. As the air...Ch. 3.1 - A ball is dropped from the top of the Empire State...Ch. 3.1 - At a time t seconds after it is thrown up in the...Ch. 3.1 - Let f(t) and g(t) give, respectively, the amount...Ch. 3.1 - A jkulhlaup is the rapid draining of a glacial...Ch. 3.1 - (a) For y = kxn, show that near any point x = a,...Ch. 3.1 - In Problems 8992, use the fact that for a power...Ch. 3.1 - In Problems 8992, use the fact that for a power...Ch. 3.1 - In Problems 8992, use the fact that for a power...Ch. 3.1 - In Problems 8992, use the fact that for a power...Ch. 3.1 - The depth, h (in mm), of the water runoff down a...Ch. 3.1 - If M is the mass of the earth and G is a constant,...Ch. 3.1 - The period, T, of a pendulum is given in terms of...Ch. 3.1 - (a) Use the formula for the area of a circle of...Ch. 3.1 - Show that for any power function f(x) = xn, we...Ch. 3.1 - Suppose W is proportional to r3. The derivative...Ch. 3.1 - Given a power function of the form f(x) = axn,...Ch. 3.1 - (a) Find the value of a making f(x) continuous at...Ch. 3.1 - Let f(x)=x3+3x22x+1 and g(x)=x3+3x22x4. (a) Show...Ch. 3.1 - Let f(x)=x43x2+1. (a) Show that f(x) is an even...Ch. 3.1 - In Problems 103104, explain what is wrong with the...Ch. 3.1 - In Problems 103104, explain what is wrong with the...Ch. 3.1 - In Problems 105107, give an example of: Two...Ch. 3.1 - In Problems 105107, give an example of: A function...Ch. 3.1 - In Problems 105107, give an example of: A function...Ch. 3.1 - In Problems 108110, is the statement true or...Ch. 3.1 - In Problems 108110, is the statement true or...Ch. 3.1 - In Problems 108110, is the statement true or...Ch. 3.1 - In Problems 111112, is the statement true or...Ch. 3.1 - In Problems 111112, is the statement true or...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 125, find the derivatives of the...Ch. 3.2 - In Exercises 2628, find formulas for f and f....Ch. 3.2 - In Exercises 2628, find formulas for f and f....Ch. 3.2 - In Exercises 2628, find formulas for f and f....Ch. 3.2 - In Problems 2939, can the functions be...Ch. 3.2 - In Problems 2939, can the functions be...Ch. 3.2 - In Problems 2939, can the functions be...Ch. 3.2 - In Problems 2939, can the functions be...Ch. 3.2 - In Problems 2939, can the functions be...Ch. 3.2 - In Problems 2939, can the functions be...Ch. 3.2 - In Problems 2939, can the functions be...Ch. 3.2 - In Problems 2939, can the functions be...Ch. 3.2 - In Problems 2939, can the functions be...Ch. 3.2 - In Problems 2939, can the functions be...Ch. 3.2 - In Problems 2939, can the functions be...Ch. 3.2 - (a) Use Figure 3.12 to rank the quantities f (1),...Ch. 3.2 - An animal population is given by P(t) =...Ch. 3.2 - With a yearly inflation rate of 2%, prices are...Ch. 3.2 - After a storm, a high-pressure system moves into...Ch. 3.2 - The value of an automobile purchased in 2014 can...Ch. 3.2 - With t in years since the start of 2014, worldwide...Ch. 3.2 - Food bank usage in Britain has grown dramatically...Ch. 3.2 - In 2012, the population of Mexico was 115 million...Ch. 3.2 - Some antique furniture increased very rapidly in...Ch. 3.2 - Find the quadratic polynomial g(x)=ax2+bx+c which...Ch. 3.2 - In Problems 5051, explain what is wrong with the...Ch. 3.2 - In Problems 5051, explain what is wrong with the...Ch. 3.2 - In Problems 5253, give an example of: An...Ch. 3.2 - In Problems 5253, give an example of: A function f...Ch. 3.2 - Are the statements in Problems 5456 true or false?...Ch. 3.2 - Are the statements in Problems 5456 true or false?...Ch. 3.2 - Are the statements in Problems 5456 true or false?...Ch. 3.3 - If f(x)=x2(x3+5), find f(x) two ways: by using the...Ch. 3.3 - If f(x)=2x3x, find f(x) two ways: by using the...Ch. 3.3 - For Exercises 330, find the derivative. 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It may be...Ch. 3.3 - In Problems 3133, use Figure 3.14 and the product...Ch. 3.3 - In Problems 3133, use Figure 3.14 and the product...Ch. 3.3 - In Problems 3133, use Figure 3.14 and the product...Ch. 3.3 - For Problems 3439, let h(x)=f(x)g(x), and...Ch. 3.3 - For Problems 3439, let h(x)=f(x)g(x), and...Ch. 3.3 - For Problems 3439, let h(x)=f(x)g(x), and...Ch. 3.3 - For Problems 3439, let h(x)=f(x)g(x), and...Ch. 3.3 - For Problems 3439, let h(x)=f(x)g(x), and...Ch. 3.3 - For Problems 3439, let h(x)=f(x)g(x), and...Ch. 3.3 - Differentiate f(t)=et by writing it as f(t)=1et.Ch. 3.3 - Differentiate f(x)=e2x by writing it as f(x)=exex.Ch. 3.3 - Differentiate f(x)=e3x by writing it as f(x)=exe2x...Ch. 3.3 - For what intervals is f(x)=xex concave up?Ch. 3.3 - For what intervals is g(x)=1x2+1 concave down?Ch. 3.3 - Find the equation of the tangent line to the graph...Ch. 3.3 - Find the equation of the tangent line to the graph...Ch. 3.3 - Find the equation of the tangent line to the graph...Ch. 3.3 - In Problems 4851, the functions f(x), g(x), and...Ch. 3.3 - In Problems 4851, the functions f(x), g(x), and...Ch. 3.3 - In Problems 4851, the functions f(x), g(x), and...Ch. 3.3 - In Problems 4851, the functions f(x), g(x), and...Ch. 3.3 - The differentiable functions f and g have the...Ch. 3.3 - If H (3) = 1, H (3) = 3, F(3) = 5, F (3) = 4,...Ch. 3.3 - Find the slope of the line tangent to...Ch. 3.3 - Find a possible formula for a function y = f(x)...Ch. 3.3 - The density of veins on leaves tells us about a...Ch. 3.3 - The quantity, q, of a skateboard sold depends on...Ch. 3.3 - A museum has decided to sell one of its paintings...Ch. 3.3 - Let f() be the gas consumption (in liters/km) of a...Ch. 3.3 - The manager of a political campaign uses two...Ch. 3.3 - A mutual fund holds N(t) million shares of a...Ch. 3.3 - A patients total cholesterol level, T (t), and...Ch. 3.3 - For Problems 6366, use the graphs of h(x) and g(x)...Ch. 3.3 - For Problems 6366, use the graphs of h(x) and g(x)...Ch. 3.3 - For Problems 6366, use the graphs of h(x) and g(x)...Ch. 3.3 - For Problems 6366, use the graphs of h(x) and g(x)...Ch. 3.3 - In Problems 6769, explain what is wrong with the...Ch. 3.3 - In Problems 6769, explain what is wrong with the...Ch. 3.3 - In Problems 6769, explain what is wrong with the...Ch. 3.3 - In Problems 7071, give an example of: A function...Ch. 3.3 - In Problems 7071, give an example of: A function...Ch. 3.3 - Are the statements in Problems 7274 true or false?...Ch. 3.3 - Are the statements in Problems 7274 true or false?...Ch. 3.3 - Are the statements in Problems 7274 true or false?...Ch. 3.3 - Which of the following would be a counterexample...Ch. 3.3 - In Problems 7679, let f(x) = g(x)h(x), with g (2)...Ch. 3.3 - In Problems 7679, let f(x) = g(x)h(x), with g(2) =...Ch. 3.3 - In Problems 7679, let f(x) = g(x)h(x), with g (2)...Ch. 3.3 - In Problems 7679, let f(x) = g(x)h(x), with g (2)...Ch. 3.4 - In Exercises 157, find the derivatives. 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Assume...Ch. 3.4 - In Problems 5861, use Figure 3.16 and the chain...Ch. 3.4 - In Problems 5861, use Figure 3.16 and the chain...Ch. 3.4 - In Problems 5861, use Figure 3.16 and the chain...Ch. 3.4 - In Problems 5861, use Figure 3.16 and the chain...Ch. 3.4 - In Problems 6265, use Figure 3.17 to evaluate the...Ch. 3.4 - In Problems 6265, use Figure 3.17 to evaluate the...Ch. 3.4 - In Problems 6265, use Figure 3.17 to evaluate the...Ch. 3.4 - In Problems 6265, use Figure 3.17 to evaluate the...Ch. 3.4 - Find the equation of the tangent line to f(x) = (x...Ch. 3.4 - Find the equation of the line tangent to y = f(x)...Ch. 3.4 - Find the equation of the line tangent to f(t) =...Ch. 3.4 - For what values of x is the graph of y=ex2 concave...Ch. 3.4 - For what intervals is f(x)=xex concave down?Ch. 3.4 - Suppose f(x)=(2x+1)10(3x1)7. Find a formula for f...Ch. 3.4 - Find a possible formula for a function m(x) such...Ch. 3.4 - Given F(2) = 1, F(2) = 5, F(4) = 3, F(4) = 7 and...Ch. 3.4 - Given f(x) with f(1) = 2 and f (1) = 5, find (a)...Ch. 3.4 - Given f(x) with f(2) = 7 and f (2) = 3 and f (4) =...Ch. 3.4 - A particle is moving on the x-axis, where x is in...Ch. 3.4 - A fish population is approximated by P(t)=10e0.6t,...Ch. 3.4 - The worlds population8 is about f(t)=7.17e0.011t...Ch. 3.4 - Fourth-quarter net sales for the Hershey Company9,...Ch. 3.4 - For t in years since 2010, daily oil consumption...Ch. 3.4 - The balance in a bank account t years after money...Ch. 3.4 - For t 0 in minutes, the temperature, H, of a pot...Ch. 3.4 - A yam is put in a hot oven, maintained at a...Ch. 3.4 - The 2010 census12 determined the population of the...Ch. 3.4 - If you invest P dollars in a bank account at an...Ch. 3.4 - The theory of relativity predicts that an object...Ch. 3.4 - Since the 1950s, the carbon dioxide concentration...Ch. 3.4 - Instruments on a plane measure the distance...Ch. 3.4 - For Problems 8992, if f(x) has a positive slope...Ch. 3.4 - For Problems 8992, if f(x) has a positive slope...Ch. 3.4 - For Problems 8992, if f(x) has a positive slope...Ch. 3.4 - For Problems 8992, if f(x) has a positive slope...Ch. 3.4 - In Problems 9395, explain what is wrong with the...Ch. 3.4 - In Problems 9395, explain what is wrong with the...Ch. 3.4 - In Problems 9395, explain what is wrong with the...Ch. 3.4 - In Problems 9697, give an example of: A function...Ch. 3.4 - In Problems 9697, give an example of: A function...Ch. 3.4 - Are the statements in Problems 98101 true or...Ch. 3.4 - Are the statements in Problems 98101 true or...Ch. 3.4 - Are the statements in Problems 98101 true or...Ch. 3.4 - Are the statements in Problems 98101 true or...Ch. 3.5 - Construct a table of values for cos x, x = 0, 0.1,...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - 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In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - Prob. 35ECh. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Exercises 249, find the derivatives of the...Ch. 3.5 - In Problems 5052, find formulas for f and f....Ch. 3.5 - In Problems 5052, find formulas for f and f....Ch. 3.5 - In Problems 5052, find formulas for f and f....Ch. 3.5 - Is the graph of y = sin(x4) increasing or...Ch. 3.5 - Find the line tangent to f(t)=3sin(2t)+5 at the...Ch. 3.5 - Find the line tangent to f(x)=3x+cos(5x) at the...Ch. 3.5 - Find the line tangent to f(t)=8+sin(3t) at the...Ch. 3.5 - Find the 50th derivative of y = cos x.Ch. 3.5 - Find a function F(x) satisfying F (x) = sin(4x).Ch. 3.5 - Let f(x)=sin2x+cos2x. (a) Find f(x) using the...Ch. 3.5 - On page 42 the depth, y, in feet, of water in...Ch. 3.5 - A boat at anchor is bobbing up and down in the...Ch. 3.5 - The voltage, V, in volts, in an electrical outlet...Ch. 3.5 - An oscillating mass of m gm at the end of a spring...Ch. 3.5 - With t in years, the population of a herd of deer...Ch. 3.5 - An environmentalist reports that the depth of the...Ch. 3.5 - Normal human body temperature fluctuates with a...Ch. 3.5 - In Problems 6770, find and interpret the value of...Ch. 3.5 - In Problems 6770, find and interpret the value of...Ch. 3.5 - In Problems 6770, find and interpret the value of...Ch. 3.5 - Prob. 70ECh. 3.5 - A rubber duck bounces up and down in a pool after...Ch. 3.5 - The metal bar of length l in Figure 3.23 has one...Ch. 3.5 - Let f(x) = ex sin x. (a) Find the derivative f(x)....Ch. 3.5 - In Problems 7475, explain what is wrong with the...Ch. 3.5 - In Problems 7475, explain what is wrong with the...Ch. 3.5 - In Problems 7677, give an example of: A...Ch. 3.5 - In Problems 7677, give an example of: A function...Ch. 3.5 - Are the statements in Problems 7880 true or false?...Ch. 3.5 - Are the statements in Problems 7880 true or false?...Ch. 3.5 - Are the statements in Problems 7880 true or false?...Ch. 3.6 - For Exercises 141, find the derivative. It may be...Ch. 3.6 - For Exercises 141, find the derivative. It may be...Ch. 3.6 - For Exercises 141, find the derivative. It may be...Ch. 3.6 - For Exercises 141, find the derivative. It may be...Ch. 3.6 - For Exercises 141, find the derivative. It may be...Ch. 3.6 - For Exercises 141, find the derivative. It may be...Ch. 3.6 - For Exercises 141, find the derivative. It may be...Ch. 3.6 - For Exercises 141, find the derivative. 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It may be...Ch. 3.6 - For Exercises 141, find the derivative. It may be...Ch. 3.6 - Let f(x) = ln(3x). (a) Find f(x) and simplify your...Ch. 3.6 - On what intervals is ln(x2 + 1) concave up?Ch. 3.6 - Use the chain rule to obtain the formula for...Ch. 3.6 - Using the chain rule, find ddx(logx). (Recall log...Ch. 3.6 - To compare the acidity of different solutions,...Ch. 3.6 - The number of years, T, it takes an investment of...Ch. 3.6 - A firm estimates that the total revenue, R, in...Ch. 3.6 - Average leaf width, w (in mm), in tropical...Ch. 3.6 - (a) Find the equation of the tangent line to y =...Ch. 3.6 - (a) For x 0, find and simplify the derivative of...Ch. 3.6 - (a) Given that f(x) = x3, find f(2). (b) Find...Ch. 3.6 - (a) For f(x) = 2x5 + 3x3 + x, find f(x). (b) How...Ch. 3.6 - Imagine you are zooming in on the graph of each of...Ch. 3.6 - In Problems 5558, use Figure 3.25 to find a point...Ch. 3.6 - In Problems 5558, use Figure 3.25 to find a point...Ch. 3.6 - In Problems 5558, use Figure 3.25 to find a point...Ch. 3.6 - In Problems 5558, use Figure 3.25 to find a point...Ch. 3.6 - In Problems 5961, use Figure 3.26 to estimate the...Ch. 3.6 - In Problems 5961, use Figure 3.26 to estimate the...Ch. 3.6 - In Problems 5961, use Figure 3.26 to estimate the...Ch. 3.6 - In Problems 6264, use Figure 3.27 to calculate the...Ch. 3.6 - In Problems 6264, use Figure 3.27 to calculate the...Ch. 3.6 - In Problems 6264, use Figure 3.27 to calculate the...Ch. 3.6 - Use the table and the fact that f(x) is invertible...Ch. 3.6 - At a particular location, f(p) is the number of...Ch. 3.6 - Let P = f(t) give the US population18 in millions...Ch. 3.6 - Figure 3.28 shows the number of motor vehicles,19...Ch. 3.6 - Using Figure 3.29, where f(2) = 2.1, f(4) = 3.0,...Ch. 3.6 - Figure 3.30 shows an invertible function f(x). (a)...Ch. 3.6 - An invertible function f(x) has values in the...Ch. 3.6 - Assuming g(x) 0, use the chain rule to explain...Ch. 3.6 - If f is continuous, invertible, and defined for...Ch. 3.6 - Suppose f(x) is an invertible function that is...Ch. 3.6 - In Problems 7577, explain what is wrong with the...Ch. 3.6 - In Problems 7577, explain what is wrong with the...Ch. 3.6 - In Problems 7577, explain what is wrong with the...Ch. 3.6 - In Problems 7881, give an example of: A function...Ch. 3.6 - In Problems 7881, give an example of: A function...Ch. 3.6 - In Problems 7881, give an example of: A function...Ch. 3.6 - In Problems 7881, give an example of: A function f...Ch. 3.6 - Are the statements in Problems 8283 true or false?...Ch. 3.6 - Are the statements in Problems 8283 true or false?...Ch. 3.7 - Let f(x) = 7. Using the definition of the...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dy/dx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - For Exercises 121, find dydx. Assume a, b, c are...Ch. 3.7 - In Exercises 2225, find the slope of the tangent...Ch. 3.7 - In Exercises 2225, find the slope of the tangent...Ch. 3.7 - In Exercises 2225, find the slope of the tangent...Ch. 3.7 - In Exercises 2225, find the slope of the tangent...Ch. 3.7 - For Exercises 2630, find the equations of the...Ch. 3.7 - For Exercises 2630, find the equations of the...Ch. 3.7 - For Exercises 2630, find the equations of the...Ch. 3.7 - For Exercises 2630, find the equations of the...Ch. 3.7 - For Exercises 2630, find the equations of the...Ch. 3.7 - (a) Find dydx given that x2 + y2 4x + 7y = 15....Ch. 3.7 - (a) Find the slope of the tangent line to the...Ch. 3.7 - (a) Find all points on y2 + xy + x2 = 1 with x =...Ch. 3.7 - Find the equations of the tangent lines at x = 2...Ch. 3.7 - (a) Find the equations of the tangent lines to the...Ch. 3.7 - Find the equation of the tangent line to the curve...Ch. 3.7 - If y = arcsin x then x = sin y. Use implicit...Ch. 3.7 - Show that the power rule for derivatives applies...Ch. 3.7 - At pressure P atmospheres, a certain fraction f of...Ch. 3.7 - For constants a, b, n, R, Van der Waals equation...Ch. 3.7 - In Problems 4142, explain what is wrong with the...Ch. 3.7 - In Problems 4142, explain what is wrong with the...Ch. 3.7 - In Problems 4344, give an example of: A formula...Ch. 3.7 - In Problems 4344, give an example of: A curve that...Ch. 3.7 - True or false? Explain your answer: If y satisfies...Ch. 3.8 - In Exercises 111, find the derivative of the...Ch. 3.8 - In Exercises 111, find the derivative of the...Ch. 3.8 - In Exercises 111, find the derivative of the...Ch. 3.8 - In Exercises 111, find the derivative of the...Ch. 3.8 - In Exercises 111, find the derivative of the...Ch. 3.8 - In Exercises 111, find the derivative of the...Ch. 3.8 - In Exercises 111, find the derivative of the...Ch. 3.8 - In Exercises 111, find the derivative of the...Ch. 3.8 - In Exercises 111, find the derivative of the...Ch. 3.8 - In Exercises 111, find the derivative of the...Ch. 3.8 - In Exercises 111, find the derivative of the...Ch. 3.8 - Show that d(sinh x)dx = cosh x.Ch. 3.8 - Show that d(sinh x)/dx = cosh xCh. 3.8 - Show that sinh(x) = sinh(x).Ch. 3.8 - In Exercises 1516, simplify the expressions....Ch. 3.8 - In Exercises 1516, simplify the expressions....Ch. 3.8 - Describe and explain the behavior of sinh x as x ...Ch. 3.8 - If x = cosh t and y = sinh t, explain why the...Ch. 3.8 - Is there an identity analogous to sin(2x) = 2 sin...Ch. 3.8 - Is there an identity analogous to cos(2x) = cos2 x...Ch. 3.8 - Prove the identities in Problems 2122. cosh(A + B)...Ch. 3.8 - Prove the identities in Problems 2122. sinh(A + B)...Ch. 3.8 - In Problems 2326, find the limit of the function...Ch. 3.8 - In Problems 2326, find the limit of the function...Ch. 3.8 - In Problems 2326, find the limit of the function...Ch. 3.8 - In Problems 2326, find the limit of the function...Ch. 3.8 - For what values of k is limxe3x cosh kx finite?Ch. 3.8 - For what values of k is limxsinhkxcosh2x finite?Ch. 3.8 - The cable between the two towers of a power line...Ch. 3.8 - The Saint Louis Arch can be approximated by using...Ch. 3.8 - (a) Using a calculator or computer, sketch the...Ch. 3.8 - The following problem is a generalization of...Ch. 3.8 - (a) Find tanh 0. (b) For what values of x is tanh...Ch. 3.8 - In Problems 3437, explain what is wrong with the...Ch. 3.8 - In Problems 3437, explain what is wrong with the...Ch. 3.8 - In Problems 3437, explain what is wrong with the...Ch. 3.8 - In Problems 3437, explain what is wrong with the...Ch. 3.8 - In Problems 3840, give an example of: A hyperbolic...Ch. 3.8 - In Problems 3840, give an example of: A value of k...Ch. 3.8 - In Problems 3840, give an example of: A function...Ch. 3.8 - Are the statements in Problems 4145 true or false?...Ch. 3.8 - Are the statements in Problems 4145 true or false?...Ch. 3.8 - Are the statements in Problems 4145 true or false?...Ch. 3.8 - Are the statements in Problems 4145 true or false?...Ch. 3.8 - Are the statements in Problems 4145 true or false?...Ch. 3.9 - Find the tangent line approximation for 1+x near x...Ch. 3.9 - What is the tangent line approximation to ex near...Ch. 3.9 - Find the tangent line approximation to 1x near x =...Ch. 3.9 - Find the local linearization of f(x) = x2 near x =...Ch. 3.9 - What is the local linearization of ex2 near x = 1?Ch. 3.9 - Show that 1 x2 is the tangent line approximation...Ch. 3.9 - Show that ex 1 x near x = 0.Ch. 3.9 - Local linearization gives values too small for the...Ch. 3.9 - Using a graph like Figure 3.37, estimate to one...Ch. 3.9 - For x near 0, local linearization gives ex 1 + x....Ch. 3.9 - (a) Find the best linear approximation, L(x), to...Ch. 3.9 - (a) Find the tangent line approximation to cos x...Ch. 3.9 - Suppose f(x) has f(50) = 99.5 and f50) = 0.2. (a)...Ch. 3.9 - The graphs in Figure 3.39 have the same window and...Ch. 3.9 - (a) Graph f(x) = x3 3x2 + 3x + 1. (b) Find and...Ch. 3.9 - (a) Show that 1+kx is the local linearization of...Ch. 3.9 - Figure 3.40 shows f(x) and its local linearization...Ch. 3.9 - In Problems 1819, the equation has a solution near...Ch. 3.9 - In Problems 1819, the equation has a solution near...Ch. 3.9 - (a) Given that f(7) = 13 and f(7) = 0.38, estimate...Ch. 3.9 - A function g has g(3) = 7 and g(3.001) g(3) =...Ch. 3.9 - (a) Explain why the following equation has a...Ch. 3.9 - The speed of sound in dry air is...Ch. 3.9 - Live phytoplankton of diameter x micrometers sink...Ch. 3.9 - The generation time for an organism is the time...Ch. 3.9 - Air pressure at sea level is 30 inches of mercury....Ch. 3.9 - On October 7, 2010, the Wall Street Journal22...Ch. 3.9 - Table 3.9 shows the water stored S(t), in...Ch. 3.9 - Small water bugs swim in groups as protection...Ch. 3.9 - If C (in units of 104 molar) is the concentration...Ch. 3.9 - Writing g for the acceleration due to gravity, the...Ch. 3.9 - Suppose now the length of the pendulum in Problem...Ch. 3.9 - Suppose f has a continuous positive second...Ch. 3.9 - Suppose f(x) is a differentiable decreasing...Ch. 3.9 - Problems 3537 investigate the motion of a...Ch. 3.9 - Problems 3537 investigate the motion of a...Ch. 3.9 - Problems 3537 investigate the motion of a...Ch. 3.9 - In Problems 3840, find the local linearization of...Ch. 3.9 - In Problems 3840, find the local linearization of...Ch. 3.9 - In Problems 3840, find the local linearization of...Ch. 3.9 - In Problems 4145, find a formula for the error...Ch. 3.9 - In Problems 4145, find a formula for the error...Ch. 3.9 - In Problems 4145, find a formula for the error...Ch. 3.9 - In Problems 4145, find a formula for the error...Ch. 3.9 - In Problems 4145, find a formula for the error...Ch. 3.9 - Multiply the local linearization of ex near x = 0...Ch. 3.9 - (a) Show that 1 x is the local linearization of...Ch. 3.9 - From the local linearizations of ex and sin x near...Ch. 3.9 - Use local linearization to derive the product...Ch. 3.9 - Derive the chain rule using local linearization....Ch. 3.9 - Consider a function f and a point a. Suppose there...Ch. 3.9 - Consider the graph of f(x) = x2 near x = 1. Find...Ch. 3.9 - In Problems 5354, explain what is wrong with the...Ch. 3.9 - In Problems 5354, explain what is wrong with the...Ch. 3.9 - In Problems 5557, give an example of: Two...Ch. 3.9 - In Problems 5557, give an example of: A...Ch. 3.9 - In Problems 5557, give an example of: A function...Ch. 3.9 - Let f be a differentiable function and let L be...Ch. 3.10 - Let f(x) = 7. Using the definition of the...Ch. 3.10 - In Exercises 15, decide if the statements are true...Ch. 3.10 - In Exercises 15, decide if the statements are true...Ch. 3.10 - In Exercises 15, decide if the statements are true...Ch. 3.10 - In Exercises 15, decide if the statements are true...Ch. 3.10 - Do the functions graphed in Exercises 69 appear to...Ch. 3.10 - Do the functions graphed in Exercises 69 appear to...Ch. 3.10 - Do the functions graphed in Exercises 69 appear to...Ch. 3.10 - Do the functions graphed in Exercises 69 appear to...Ch. 3.10 - Applying the Mean Value Theorem with a = 2, b = 7...Ch. 3.10 - Applying the Mean Value Theorem with a = 3, b = 13...Ch. 3.10 - Let p(x) = x5 +8x4 30x3 +30x2 31x+22. What is the...Ch. 3.10 - Let p(x) be a seventh-degree polynomial with 7...Ch. 3.10 - Use the Racetrack Principle and the fact that sin...Ch. 3.10 - Use the Racetrack Principle to show that ln x x1.Ch. 3.10 - Use the fact that ln x and ex are inverse...Ch. 3.10 - State a Decreasing Function Theorem, analogous to...Ch. 3.10 - Dominic drove from Phoenix to Tucson on Interstate...Ch. 3.10 - In Problems 1922, use one of the theorems in this...Ch. 3.10 - In Problems 1922, use one of the theorems in this...Ch. 3.10 - In Problems 1922, use one of the theorems in this...Ch. 3.10 - In Problems 1922, use one of the theorems in this...Ch. 3.10 - The position of a particle on the x-axis is given...Ch. 3.10 - Suppose that g and are continuous on [a, b] and...Ch. 3.10 - Deduce the Constant Function Theorem from the...Ch. 3.10 - Prove that if f (x) = g(x) for all x in (a, b),...Ch. 3.10 - Suppose that f(x) = f(x) for all x. Prove that...Ch. 3.10 - Suppose that f is continuous on [a, b] and...Ch. 3.10 - Suppose that f(x) 0 for all x in (a, b). We will...Ch. 3.10 - In Problems 3032, explain what is wrong with the...Ch. 3.10 - In Problems 3032, explain what is wrong with the...Ch. 3.10 - In Problems 3032, explain what is wrong with the...Ch. 3.10 - In Problems 3337, give an example of: An interval...Ch. 3.10 - In Problems 3337, give an example of: An interval...Ch. 3.10 - In Problems 3337, give an example of: A continuous...Ch. 3.10 - In Problems 3337, give an example of: A function f...Ch. 3.10 - In Problems 3337, give an example of: A function...Ch. 3.10 - Are the statements in Problems 3841 true or false...Ch. 3.10 - Are the statements in Problems 3841 true or false...Ch. 3.10 - Are the statements in Problems 3841 true or false...Ch. 3.10 - Are the statements in Problems 3841 true or false...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - 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Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - Find derivatives for the functions in Exercises...Ch. 3 - For Exercises 7475, assume that y is a...Ch. 3 - For Exercises 7475, assume that y is a...Ch. 3 - Find the slope of the curve x2 + 3y2 = 7 at (2,...Ch. 3 - Assume y is a differentiable function of x and...Ch. 3 - Find the equations for the lines tangent to the...Ch. 3 - If f(t) = 2t3 4t2 + 3t 1, find f(t) and f(t).Ch. 3 - If f(x)=138x+2x2 and f(r) = 4, find r.Ch. 3 - If f(x) = x3 6x2 15x+20, find analytically all...Ch. 3 - For Problems 8287, use Figure 3.1. Figure 3.1 Let...Ch. 3 - For Problems 8287, use Figure 3.1. Figure 3.1 Let...Ch. 3 - For Problems 8287, use Figure 3.1. Figure 3.1 Let...Ch. 3 - For Problems 8287, use Figure 3.1. Figure 3.1...Ch. 3 - For Problems 8287, use Figure 3.1. Figure 3.1 Let...Ch. 3 - For Problems 8287, use Figure 3.1. Figure 3.1 Find...Ch. 3 - For Problems 8891, let (x) = f(g(x)) and k(x) =...Ch. 3 - For Problems 8891, let (x) = f(g(x)) and k(x) =...Ch. 3 - For Problems 8891, let (x) = f(g(x)) and k(x) =...Ch. 3 - For Problems 8891, let (x) = f(g(x)) and k(x) =...Ch. 3 - Using the information in the table about f and g,...Ch. 3 - Given: r(2) = 4, s(2) = 1, s(4) = 2, r(2) = 1,...Ch. 3 - If g(2) = 3 and g(2) = 4, find f (2) for the...Ch. 3 - For parts (a)(f) of Problem 94, determine the...Ch. 3 - Imagine you are zooming in on the graphs of the...Ch. 3 - The graphs of sin x and cos x intersect once...Ch. 3 - In Problems 9899, show that the curves meet at...Ch. 3 - In Problems 9899, show that the curves meet at...Ch. 3 - For some constant b and x 0, let y = x ln xbx....Ch. 3 - In Problems 101103, find the limit as x ....Ch. 3 - In Problems 101103, find the limit as x ....Ch. 3 - In Problems 101103, find the limit as x ....Ch. 3 - Consider the function f(x)=x. (a) Find and sketch...Ch. 3 - Figure 3.3 shows the tangent line approximation to...Ch. 3 - Global temperatures have been rising, on average,...Ch. 3 - The population of Hungary2 has been modeled by P =...Ch. 3 - Prob. 108RECh. 3 - The distance, s, of a moving body from a fixed...Ch. 3 - At any time, t, a population, P(t), is growing at...Ch. 3 - An object is oscillating at the end of a spring....Ch. 3 - The total number of people, N, who have contracted...Ch. 3 - The world population3 was 7.17 billion in July of...Ch. 3 - The acceleration due to gravity, g, is given by...Ch. 3 - Given that f and g are differentiable everywhere,...Ch. 3 - Prob. 116RECh. 3 - A particle is moving on the x-axis. It has...Ch. 3 - If f is decreasing and f(20) = 10, which of the...Ch. 3 - Find the nth derivative of the following...Ch. 3 - The derivative f gives the (absolute) rate of...Ch. 3 - The relative rate of change of a function f is...
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY