question given $f(x)=3x^{3}-x^{2}-2x$, find the equation of the tangent line of $f$ at the point where $x =…
question given $f(x)=3x^{3}-x^{2}-2x$, find the equation of the tangent line of $f$ at the point where $x = 1$.
Answer
Explanation:
Step1: Find the derivative of the function
Using the power - rule $(x^n)'=nx^{n - 1}$, if $f(x)=3x^{3}-x^{2}-2x$, then $f'(x)=9x^{2}-2x - 2$.
Step2: Find the slope of the tangent line at $x = 1$
Substitute $x = 1$ into $f'(x)$. So $m=f'(1)=9(1)^{2}-2(1)-2=9 - 2 - 2=5$.
Step3: Find the y - coordinate of the point on the function at $x = 1$
Substitute $x = 1$ into $f(x)$. So $y=f(1)=3(1)^{3}-(1)^{2}-2(1)=3 - 1 - 2=0$.
Step4: Use the point - slope form of a line
The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(1,0)$ and $m = 5$. So $y-0=5(x - 1)$, which simplifies to $y = 5x-5$.
Answer:
$y = 5x-5$