Linear approximation and (the second derivative) Draw the

Chapter 4, Problem 47E

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QUESTION:

Draw the graph of a function f such that \(f(1)=f^{\prime}(1)=f^{\prime \prime}(1)=1\). Draw the linear approximation to the function at the point (1, 1). Now draw the graph of another function g such that \(g(1)=g^{\prime}(1)=1\) and \(g^{\prime \prime}(1)=10\0. (It is not possible to represent the second derivative exactly, but your graphs should reflect the fact that \(f^{\prime \prime}(1)\) is relatively small and \(g^{\prime \prime}(1)\) is relatively large.) Now, suppose that linear approximations are used to approximate f(1.1) and g(1.1).

a. Which function has the more accurate linear approximation near x = 1 and why?

b. Explain why the error in the linear approximation to f near a point a is proportional to the magnitude of \(f^{\prime \prime}(a)\).

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QUESTION:

Draw the graph of a function f such that \(f(1)=f^{\prime}(1)=f^{\prime \prime}(1)=1\). Draw the linear approximation to the function at the point (1, 1). Now draw the graph of another function g such that \(g(1)=g^{\prime}(1)=1\) and \(g^{\prime \prime}(1)=10\0. (It is not possible to represent the second derivative exactly, but your graphs should reflect the fact that \(f^{\prime \prime}(1)\) is relatively small and \(g^{\prime \prime}(1)\) is relatively large.) Now, suppose that linear approximations are used to approximate f(1.1) and g(1.1).

a. Which function has the more accurate linear approximation near x = 1 and why?

b. Explain why the error in the linear approximation to f near a point a is proportional to the magnitude of \(f^{\prime \prime}(a)\).

ANSWER:

SOLUTION STEP 1 Here it is given that f(1)(1) = (1) = 1 Thus we can understand that a=1 .The linear approximation formula is given by L(x) = f(a)+f(a)(xa) L(x) = 1+1(x1) = 1+x1 = x Thus we have L(x) = x STEP 2 The graph will be Here the black line is the actual curve and the other line is the linear approximation.since f"(1) = 1 is not very large,the change in slope will be less gradual. STEP 3 Here it is given that g(1) = g(1) = 1 and g"(1) = 10 Thus we can u

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