Question

Find the slope of the tangent line to the graph of fat the given point.$f\left(x\right)=3x+4,at(1,7)$.

Short answer

The slope of the tangent line is3 .

Step-by-step solution

Step 1. Given information.

The function here given is,

$\begin{array}{l}f\left(x\right)=3x+4\\ po\mathrm{int}=(1,7)\end{array}$

Step 2. Formula used.

The tangent line to the curve$y=f\left(x\right)$at the point$P(a,f(a\left)\right)$is the line through p with slope,

$m=\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$.

provided that this limit exists.

Step 3. Finding the slope of tangent line at given point.

Let$f\left(x\right)=3x+4$,then the slope of the tangent line at$(1,7)$is,

$\begin{array}{c}m=\underset{x\to 3}{\lim }\frac{f\left(x\right)-f\left(3\right)}{x-3}\\ =\underset{x\to 3}{\lim }\frac{(3x+4)-(3\cdot 3+4)}{x-3}\\ =\underset{x\to 3}{\lim }\frac{3x+4-9-4}{x-3}\\ =\underset{x\to 3}{\lim }\frac{3x-9}{x-3}\\ =\underset{x\to 3}{\lim }\frac{3\cancel{(x-3)}}{\cancel{(x-3)}}\\ =3\end{array}$

Therefore, the slope of the tangent line is3 .