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

a. use the definition m_tan = 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)=2 + 4x²; p(0,2) a. m_tan = 0 (type an integer or a fraction.) b. the equation of the tangent line at p is 2. (type an equation.)

a. use the definition m_tan = 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)=2 + 4x²; p(0,2) a. m_tan = 0 (type an integer or a fraction.) b. the equation of the tangent line at p is 2. (type an equation.)

Answer

Answer:

a. $m_{tan}=0$ b. $y = 2$

Explanation:

Step1: Identify function and point

$f(x)=2 + 4x^{2}$, $a = 0$, $f(a)=2$

Step2: Calculate $f(a + h)$

$f(0+h)=2 + 4h^{2}$

Step3: Find the slope $m_{tan}$

$m_{tan}=\lim_{h\rightarrow0}\frac{f(a + h)-f(a)}{h}=\lim_{h\rightarrow0}\frac{2 + 4h^{2}-2}{h}=\lim_{h\rightarrow0}\frac{4h^{2}}{h}=\lim_{h\rightarrow0}4h = 0$

Step4: Find the equation of the tangent line

The point - slope form of a line is $y - y_{1}=m(x - x_{1})$. Here $x_{1}=0,y_{1}=2,m = 0$. So $y-2=0(x - 0)$, which simplifies to $y = 2$.