Find the slope of the line tangent to the graph of f at point P.
Calculate f(a+h) and f(a) for a=0.
We have f(a+h)=f(0+h)=7+4(h)2=4h2+7 and f(a)=f(0)=7.
Use the limit definition to find the slope mtan.
Using the limit definition, we find:
mtan=h→0limhf(a+h)−f(a)=h→0limh(4h2+7)−7=h→0limh4h2=h→0lim4h=0.
The slope of the tangent line is 0.
Determine an equation of the tangent line at point P.
Use the point-slope form of the line equation.
The point-slope form is given by y−y1=m(x−x1). Here, m=0 and (x1,y1)=(0,7). Thus, the equation becomes:
y−7=0(x−0)⟹y=7.
The equation of the tangent line is y=7.
The slope of the tangent line is 0.
The equation of the tangent line is y=7.