question find the equation of the line tangent to the graph of f(x)=-x^3 - 5x^2 + 2 at x=-1. provide your…
question find the equation of the line tangent to the graph of f(x)=-x^3 - 5x^2 + 2 at x=-1. provide your answer below: y=
Answer
Explanation:
Step1: Find the derivative of the function
The derivative of $f(x)=-x^{3}-5x^{2}+2$ using the power - rule $(x^n)' = nx^{n - 1}$ is $f'(x)=-3x^{2}-10x$.
Step2: Find the slope of the tangent line
Substitute $x = - 1$ into $f'(x)$. So $m=f'(-1)=-3(-1)^{2}-10(-1)=-3 + 10=7$.
Step3: Find the y - coordinate of the point of tangency
Substitute $x=-1$ into $f(x)$. So $y=f(-1)=-(-1)^{3}-5(-1)^{2}+2=1 - 5 + 2=-2$.
Step4: Use the point - slope form of a line
The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(-1,-2)$ and $m = 7$. $y+2=7(x + 1)$.
Step5: Rewrite in slope - intercept form
Expand and simplify: $y+2=7x+7$, then $y=7x + 5$.
Answer:
$y = 7x+5$