a. use the definition mtan = lim h→0 f(a + h)-f(a)/h to find the slope of the line tangent to the graph of f…

a. use the definition mtan = lim h→0 f(a + h)-f(a)/h to find the slope of the line tangent to the graph of f at p. b. determine an equation of the tangent line at p. f(x)=5 + 7x²; p(0,5) a. mtan = (type an integer or a fraction.)

a. use the definition mtan = lim h→0 f(a + h)-f(a)/h to find the slope of the line tangent to the graph of f at p. b. determine an equation of the tangent line at p. f(x)=5 + 7x²; p(0,5) a. mtan = (type an integer or a fraction.)

Answer

Answer:

a. $0$ b. $y = 5$

Explanation:

Step1: Identify $a$ and $f(x)$

$a = 0$, $f(x)=5 + 7x^{2}$

Step2: Calculate $f(a + h)$ and $f(a)$

$f(a+h)=5 + 7(0 + h)^{2}=5+7h^{2}$, $f(a)=f(0)=5 + 7\times0^{2}=5$

Step3: Find the slope $m_{tan}$

[ \begin{align*} m_{tan}&=\lim_{h\rightarrow0}\frac{f(a + h)-f(a)}{h}\ &=\lim_{h\rightarrow0}\frac{(5 + 7h^{2})-5}{h}\ &=\lim_{h\rightarrow0}\frac{7h^{2}}{h}\ &=\lim_{h\rightarrow0}7h\ &=0 \end{align*} ]

Step4: Find the equation of the tangent - line

The point - slope form of a line is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(0,5)$ and $m = 0$. [ \begin{align*} y-5&=0(x - 0)\ y&=5 \end{align*} ]