3. (5 points) let ( f(7)=13 ) and ( f^{prime}(7)=-0.38 ).\na. (3 points) use the linear approximation of (…

3. (5 points) let ( f(7)=13 ) and ( f^{prime}(7)=-0.38 ).\na. (3 points) use the linear approximation of ( f(x) ) at ( x = 7 ) to estimate ( f(7.1) ).\nb. (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.

3. (5 points) let ( f(7)=13 ) and ( f^{prime}(7)=-0.38 ).\na. (3 points) use the linear approximation of ( f(x) ) at ( x = 7 ) to estimate ( f(7.1) ).\nb. (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.

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)): [ \begin{align*} L(7.1)&=f(7)+f^{\prime}(7)(7.1 - 7)\ &=13+(-0.38)\times(7.1 - 7)\ &=13-0.38\times0.1\ &=13 - 0.038\ &=12.962 \end{align*} ]

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.