Question
The graph of f′, the derivative of the function f, is shown above. If f(4)=−1, what is the approximation for f(4.5) using the line tangent to the graph of f at x=4 ?
A -4
B -1
C 2
D 6
▶️Answer/Explanation
Ans:C
An equation of the tangent line at x=a is y=f(a)+f′(a)(x−a). Here a=4, so f(a)=−1 and f′(a)=6. The value of y when x=4.5 would be an approximation to f(4.5), as follows.
f(4.5)≈f(4)+6(4.5−4)=−1+3=2
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?
A 0.6
B 3.4
C 4.2
D 4.6
E 4.64
▶️Answer/Explanation
Ans:D
Question
Let f be the function given by f(x) = 2 cos x + 1. What is the approximation for f(1.5) found by using the line tangent to the graph of f at \(x=\frac{\pi }{2}\) ?
A -2
B 1
C \(\pi -2\)
D \(4-\pi \)
▶️Answer/Explanation
Ans:C
Question
The function \(f\) is twice differentiable with \(f(2)=1, f^{\prime}(2)=4\), and \(f^{”}(2)=3\). What is the value of the approximation of \(f(1.9)\) using the line tangent to the graph of \(f\) at \(x=2\) ?
A 0.4
B 0.6
C 0.7
D 1.3
E 1.4
▶️Answer/Explanation
Ans:B
\[
f(2)=1, f^{\prime}(2)=4 f^{\prime \prime}(2)=3 x=2 f(1.9)=?
\]
The formula for the slope of a tangent line to a curve \(y=f(x)\) at the point \((a, f(a))\) is:
\[
\begin{aligned}
m & =\frac{y-f(a)}{x-a} \\
f^{\prime}(a) & =\frac{y-f(a)}{x-a}
\end{aligned}
\]
Rewrite the above expression for the variable \(y\) or the function \(f(x)\).
\[
\begin{aligned}
(x-a) f^{\prime}(a) & =y-f(a) \\
y-f(a) & =(x-a) f^{\prime}(a) \\
y & =(x-a) f^{\prime}(a)+f(a) \\
f(x) & =(x-a) f^{\prime}(a)+f(a) \quad[\because y=f(x)]
\end{aligned}
\]
Substitute \(a=2\) in the above function.
\[
\begin{array}{rlr}
f(x) & =(x-2) f^{\prime}(2)+f(2) & \\
& =(x-2) 4+1 \quad\left[\because f^{\prime}(2)=4 \text { and } f(2)=1\right] \\
& =4 x-8+1 \\
& =4 x-7
\end{array}
\]
Substitute \(x=1.9\) in the above function and solve for
\[
\begin{aligned}
& f(1.9) . \\
& \begin{aligned}
f(1.9) & =4(1.9)-7 \\
& =7.6-7 \\
& =0.6
\end{aligned}
\end{aligned}
\]