Question

for the function $f(x)=2x^{2}-11x - 2$, find the equation of the tangent line at $x=-1$.

Explanation:

Step1: Find the derivative of the function

The derivative of $f(x)=2x^{2}-11x - 2$ using the power - rule $(x^n)^\prime=nx^{n - 1}$ is $f^\prime(x)=4x-11$.

Step2: Find the slope of the tangent line at $x = - 1$

Substitute $x=-1$ into $f^\prime(x)$: $f^\prime(-1)=4\times(-1)-11=-4 - 11=-15$. So the slope $m=-15$.

Step3: Find the point on the function at $x = - 1$

Substitute $x = - 1$ into $f(x)$: $f(-1)=2\times(-1)^{2}-11\times(-1)-2=2 + 11-2=11$. The point is $(-1,11)$.

Step4: Use the point - slope form to find the equation of the tangent line

The point - slope form is $y - y_1=m(x - x_1)$. Substitute $m=-15$, $x_1=-1$ and $y_1 = 11$: $y-11=-15(x + 1)$. Expand to get $y-11=-15x-15$, then $y=-15x-4$.

Answer:

$y=-15x - 4$