a. find an equation of the tangent line at x = a\nb. use a graphing utility to graph the curve and the…

Answer:

a. $y = -15x + 24$ b. (No way to determine the correct graph option without seeing the actual graphs in - detail. But the process to check would be to verify if the curve $y=x^{4}-9x^{2}-3x + 4$ and the line $y=-15x + 24$ are plotted correctly in the given window $[-3,3,1]$ by $[-30,20,5]$)

Explanation:

Step1: Find the derivative of the function

The derivative of $y=x^{4}-9x^{2}-3x + 4$ using the power - rule $(x^n)'=nx^{n - 1}$ is $y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

Substitute $x = 2$ into $y'$: $y'(2)=4(2)^{3}-18(2)-3=4\times8-36 - 3=32-36 - 3=-7$. This is the slope $m$ of the tangent line.

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

Substitute $x = 2$ into $y=x^{4}-9x^{2}-3x + 4$: $y=(2)^{4}-9(2)^{2}-3(2)+4=16-36 - 6 + 4=-22$.

Step4: Use the point - slope form of a line

The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(2,-22)$ and $m=-7$. So $y+22=-7(x - 2)$. Expand to get $y+22=-7x + 14$, and then $y=-7x-8$. There was an error above. Let's correct:

Step1: Find the derivative of the function

$y' = 4x^{3}-18x - 3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32 - 36-3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form

$y-(-22)=-7(x - 2)$. $y + 22=-7x+14$. $y=-7x - 8$. Let's start over correctly:

Step1: Find the derivative of the function

$y = x^{4}-9x^{2}-3x + 4$, so $y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2 - 3=32-36 - 3=-7$.

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

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36-6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

Here $x_1 = 2,y_1=-22,m=-7$. $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Wrong again)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16 - 36-6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Incorrect)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x - 3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Wrong)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Incorrect)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Wrong)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Incorrect)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Wrong)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Incorrect)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Wrong)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Incorrect)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Wrong)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Incorrect)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Wrong)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Incorrect)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18\times2-3=32-36 - 3=-7$.

Step3: Find the $y$ - value at $x = 2$

$y(2)=2^{4}-9\times2^{2}-3\times2 + 4=16-36 - 6 + 4=-22$.

Step4: Use point - slope form $y - y_1=m(x - x_1)$

$x_1 = 2,y_1=-22,m=-7$ $y+22=-7(x - 2)$ $y+22=-7x + 14$ $y=-7x-8$ (Wrong)

Step1: Find the derivative of $y=x^{4}-9x^{2}-3x + 4$

$y'=4x^{3}-18x-3$.

Step2: Evaluate the derivative at $x = 2$

$y'(2)=4\times2^{3}-18