Determine the slope of the tangent line to f(x) = x^2 at x = 3. (2023)

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Question: Determine the slope of the tangent line to f(x) = x^2 at x = 3. (2023)

Options:

Correct Answer: 6

Exam Year: 2023

Solution:

f\'(x) = 2x; thus, f\'(3) = 2(3) = 6.

Determine the slope of the tangent line to f(x) = x^2 at x = 3. (2023)

Determine the slope of the tangent line to f(x) = x^2 at x = 3. (2023)

Step 1: Identify the function you are working with, which is f(x) = x^2.
Step 2: Find the derivative of the function, which tells us the slope of the tangent line. The derivative of f(x) = x^2 is f'(x) = 2x.
Step 3: Substitute the value of x where you want to find the slope into the derivative. Here, we substitute x = 3 into f'(x).
Step 4: Calculate f'(3) by plugging in 3 into the derivative: f'(3) = 2(3).
Step 5: Perform the multiplication: 2(3) = 6.
Step 6: Conclude that the slope of the tangent line to f(x) = x^2 at x = 3 is 6.

Derivative – The derivative of a function at a point gives the slope of the tangent line to the function at that point.
Function Evaluation – Evaluating the derivative at a specific point to find the slope.