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1. Chains Inc. is in the business of making and selling chains. Let c(t)be the number of miles of chain produced after t hours of production. Let p(c) be the profit as a function of the number of miles of chain produced and let q(t) be the profit as a function of the number of hours of production. Suppose the company can produce 3 miles of chain per hour and suppose their profit on the chains is $4000 per mile of chain. Find and interpret (use complete sentences) each of the following (include units), c' (t) p'(c), and d' (t). How does d (t) relates to p'(c)and c(t)?

1. Chains Inc. is in the business of making and selling chains. Let c(t)be the number of miles of chain produced after t hours of production. Let p(c) be the profit as a function of the number of miles of chain produced and let q(t) be the profit as a function of the number of hours of production. Suppose the company can produce 3 miles of chain per hour and suppose their profit on the chains is $4000 per mile of chain. Find and interpret (use complete sentences) each of the following (include units), c' (t) p'(c), and d' (t). How does d (t) relates to p'(c)and c(t)?

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter2: Functions
Section2.4: Average Rate Of Change Of A Function
Problem 4.2E: bThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.
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1. Chains Inc. is in the business of making and selling chains. Let c(t)be the number of miles of chain produced after thours of production. Let p(c) be the profit as a function of the number of miles of chain produced and let g(t) be the profit as a function of the number of hours of production. Suppose the company can produce 3 miles of chain per hour and suppose their profit on the chains is $4000 per mile of chain. Find and interpret (use complete sentences) each of the following (include units), c' (t) p'(c), and d' (t). How does d (t) relates to p'(c)and c(t)? 2. Use Desmos to graph the function y³ +yx²+x² − 3y² = 0) and estimate the slope of the tangent line at (-1,1). Then find using implicit differentiation and plug in x = -1 and y = 1. Compare and discuss the estimated slope with the slope you found analytically. 3. Let f(x) = (3x² + 1)2. Find f'(x) in 3 different ways by following the instructions below in parts a, b and c: a) Develop the identity f(x) then take the derivative. b) View f(x) as (3x² + 1)(3x² + 1) and use the product rule to find f'(x). c) Apply the chain rule directly to the expression f(x) = (3x² + 1)². d) Are your answers in parts a, b, c the same? Why or why not?
Transcribed Image Text:1. Chains Inc. is in the business of making and selling chains. Let c(t)be the number of miles of chain produced after thours of production. Let p(c) be the profit as a function of the number of miles of chain produced and let g(t) be the profit as a function of the number of hours of production. Suppose the company can produce 3 miles of chain per hour and suppose their profit on the chains is $4000 per mile of chain. Find and interpret (use complete sentences) each of the following (include units), c' (t) p'(c), and d' (t). How does d (t) relates to p'(c)and c(t)? 2. Use Desmos to graph the function y³ +yx²+x² − 3y² = 0) and estimate the slope of the tangent line at (-1,1). Then find using implicit differentiation and plug in x = -1 and y = 1. Compare and discuss the estimated slope with the slope you found analytically. 3. Let f(x) = (3x² + 1)2. Find f'(x) in 3 different ways by following the instructions below in parts a, b and c: a) Develop the identity f(x) then take the derivative. b) View f(x) as (3x² + 1)(3x² + 1) and use the product rule to find f'(x). c) Apply the chain rule directly to the expression f(x) = (3x² + 1)². d) Are your answers in parts a, b, c the same? Why or why not?
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