Question

let ( f ) be a differentiable function such that ( f(4)=7 ) and ( f^{prime}(4)=\frac{1}{2} ). the graph of ( f ) is concave up on the interval ( (3,5) ). which of the following is true about the approximation for ( f(3.5) ) found using the line tangent to the graph of ( f ) at ( x = 4 )?

a ( f(3.5) approx 6.9 ) and this approximation is an overestimate of the value of ( f(3.5) ).

b ( f(3.5) approx 6.9 ) and this approximation is an underestimate of the value of ( f(3.5) ).

c ( f(3.5) approx 7.1 ) and this approximation is an overestimate of the value of ( f(3.5) ).

d ( f(3.5) approx 7.1 ) and this approximation is an underestimate of the value of ( f(3.5) ).

Explanation:

Step1: Find the linear approximation formula

The linear approximation formula is \(L(x)=f(a)+f^{\prime}(a)(x - a)\). Here \(a = 4\), \(f(4)=7\), \(f^{\prime}(4)=\frac{1}{2}\), and \(x = 3.5\).

$$L(3.5)=f(4)+f^{\prime}(4)(3.5 - 4)$$

Step2: Substitute the values

Substitute \(f(4)=7\), \(f^{\prime}(4)=\frac{1}{2}\), and \(x - a=3.5 - 4=-0.5\) into the formula.

$$L(3.5)=7+\frac{1}{2}\times(- 0.5)=7 - 0.25=6.75\approx6.9$$

Step3: Analyze the concavity

Since the graph of \(y = f(x)\) is concave - up on the interval \((3,5)\), the function \(y = f(x)\) lies above its tangent line on the interval \((3,5)\).

Answer:

B. \(f(3.5)\approx6.9\) and this approximation is an underestimate of the value of \(f(3.5)\)