Question

for the function f, f(x) = 2x + 1 and f(1) = 4. what is the approximation for f(1.2) found by using the line tangent to the graph of f at x = 1?

Explanation:

Step1: Find the derivative of the function

The function is \(f(x)=2x + 1\). The derivative \(f^{\prime}(x)=\frac{d}{dx}(2x + 1)=2\).

Step2: Use the linear approximation formula

The linear approximation formula is \(L(x)=f(a)+f^{\prime}(a)(x - a)\). Here \(a = 1\), \(f(1) = 4\), \(f^{\prime}(1)=2\), and \(x=1.2\). Then \(L(1.2)=f(1)+f^{\prime}(1)(1.2 - 1)\).
Substitute the values: \(L(1.2)=4+2\times(1.2 - 1)\).

Step3: Calculate the result

First calculate \(1.2-1 = 0.2\). Then \(2\times0.2=0.4\). Finally \(4 + 0.4=4.4\).

Answer:

\(4.4\)