If an equation of the tangent line to the curve $y=f(x)$ at the point where $a=3 \text { is } y=4 x-9 \text {, }$ $\text { find } f(3) \text { and } f^{\prime}(3) \text {. }$
A) $\begin{array}{l}
f(2)=3 \
f^{\prime}(2)=3
\end{array}$
B) $\begin{array}{l}
f(2)=-3 \
f^{\prime}(2)=3
\end{array}$
C) $\begin{array}{l}
f(2)=3 \
f^{\prime}(2)=4
\end{array}$
D) $\begin{array}{l}
f(2)=9 \
f^{\prime}(2)=4
\end{array}$
E) None of these

Question

Quizplus · Accepted Answer

The equation of the tangent line is given by $y=f(a)+f^{\prime}(a)(x-a)$. Since the tangent line to the curve $y=f(x)$ at the point where $a=3$ is $y=4x-9$, we have$$f(3)+f^{\prime}(3)(x-3)=4x-9$$Substituting $x=3$ into this equation, we get$$f(3)+f^{\prime}(3)(3-3)=4(3)-9$$$$f(3)=3$$Differentiating both sides of the equation of the tangent line with respect to $x$, we get$$f^{\prime}(a)=4$$Substituting $a=3$ into this equation, we get$$f^{\prime}(3)=4$$Therefore, $f(3)=3$ and $f^{\prime}(3)=4$.

If an Equation of the Tangent Line to the Curve y=f(x)y=f(x)y=f(x)

If an Equation of the Tangent Line to the Curve $y=f(x)$