Question

a. find an equation of the tangent line at x = a
b. use a graphing utility to graph the curve and the tangent line on the same set of axes.
y=x^4 - 9x^2+5x + 4; a = 2
a. the equation of the tangent line at a = 2 is y=

Explanation:

Step1: Find the derivative of the function

The function is $y = x^{4}-9x^{2}+5x + 4$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have $y'=4x^{3}-18x + 5$.

Step2: Evaluate the derivative at $a = 2$

Substitute $x = 2$ into $y'$. So $y'(2)=4(2)^{3}-18(2)+5=4\times8-36 + 5=32-36 + 5=1$. The slope of the tangent line $m = 1$.

Step3: Find the y - coordinate when $x = 2$

Substitute $x = 2$ into $y=x^{4}-9x^{2}+5x + 4$. Then $y=(2)^{4}-9(2)^{2}+5(2)+4=16-36 + 10+4=-6$.

Step4: Use the point - slope form of a line

The point - slope form is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(2,-6)$ and $m = 1$. So $y+6=1\times(x - 2)$.

Step5: Simplify the equation

$y+6=x - 2$, which simplifies to $y=x-8$.

Answer:

$y=x - 8$