Question

1. (5 points) let ( f(7)=13 ) and ( f^{prime}(7)=-0.38 ).

a. (3 points) use the linear approximation of ( f(x) ) at ( x = 7 ) to estimate ( f(7.1) ).
b. (2 points) suppose also ( f^{prime prime}(x)>0 ) for all ( x ). does this make your answer to part (a) an under - or over - estimate? justify your answer.

Explanation:

Step1: Recall the linear approximation formula

The linear approximation formula is \(L(x)=f(a)+f^{\prime}(a)(x - a)\). Here \(a = 7\), \(x=7.1\), \(f(7)=13\) and \(f^{\prime}(7)=- 0.38\).

Step2: Substitute the values into the formula

Substitute \(a = 7\), \(x = 7.1\), \(f(7)=13\) and \(f^{\prime}(7)=-0.38\) into \(L(x)\):

$$
LATEXBLOCK0
$$

Step3: Analyze the concavity

Since \(f^{\prime\prime}(x)>0\) for all \(x\), the function \(y = f(x)\) is concave - up. The linear approximation \(L(x)\) is the equation of the tangent line at \(x = a\). For a concave - up function, the tangent line lies below the graph of the function.

Answer:

a. The linear approximation of \(f(7.1)\) is \(12.962\).
b. Since \(f^{\prime\prime}(x)>0\) (the function is concave - up), the answer in part (a) is an under - estimate. The tangent line (linear approximation) lies below the graph of the concave - up function.