Question

find the slope of the tangent line to the graph of the given function at the given value of x. find the equation of the tangent line.
$y=x^{4}-3x^{3}+5; x=2$

how would the slope of a tangent line be determined with the given information?
a. substitute 2 for x into the derivative of the function and evaluate.
b. set the derivative equal to zero and solve for x.
c. substitute values of x into the equation and solve for y. plot the resulting points to find the linear equation.
d. substitute values of y into the equation and solve for x. plot the resulting points to find the linear equation.

Explanation:

Step1: Identify the method for slope

To find the slope of the tangent line at a specific $x$-value, we use the derivative of the function, then substitute the given $x$-value into the derivative. This matches option A.

Step2: Compute the function's derivative

Apply the power rule $\frac{d}{dx}(x^n)=nx^{n-1}$ to $y=x^4 - 3x^3 + 5$.
$\frac{dy}{dx}=4x^3 - 9x^2$

Step3: Calculate the slope at $x=2$

Substitute $x=2$ into the derivative.
$\frac{dy}{dx}\bigg|_{x=2}=4(2)^3 - 9(2)^2=4(8)-9(4)=32-36=-4$

Step4: Find the $y$-value at $x=2$

Substitute $x=2$ into the original function.
$y=(2)^4 - 3(2)^3 + 5=16-24+5=-3$

Step5: Write the tangent line equation

Use the point-slope form $y-y_1=m(x-x_1)$, where $m=-4$, $x_1=2$, $y_1=-3$.
$y-(-3)=-4(x-2)$
Simplify to slope-intercept form: $y+3=-4x+8 \implies y=-4x+5$

Answer:

Multiple Choice Answer:

A. Substitute 2 for x into the derivative of the function and evaluate.

Tangent Line Results:

Slope of the tangent line: $-4$
Equation of the tangent line: $y=-4x+5$