Question

selected values of the derivative of the function g are given in the table above. it is known that g(4)=12. what is the approximation for g(4.2) found using the line tangent to the graph of g at x = 4?
a 12.44
b 12.40
c 12.36
d 11.60

Explanation:

Step1: Recall the linear approximation formula

The linear approximation formula is \(L(x)=g(a)+g^{\prime}(a)(x - a)\). Here \(a = 4\), \(x=4.2\), \(g(4)=12\), and from the table \(g^{\prime}(4)=2.2\).

Step2: Substitute values into the formula

Substitute \(a = 4\), \(x = 4.2\), \(g(4)=12\), \(g^{\prime}(4)=2.2\) into \(L(x)=g(a)+g^{\prime}(a)(x - a)\).
We get \(L(4.2)=g(4)+g^{\prime}(4)(4.2 - 4)\).
Calculate \(4.2-4 = 0.2\).
Then \(L(4.2)=12+2.2\times0.2\).

Step3: Perform the multiplication and addition

First, \(2.2\times0.2=\frac{22}{10}\times\frac{2}{10}=\frac{44}{100}=0.44\).
Then \(12 + 0.44=12.44\).

Answer:

A. 12.44