MATI Homework: 2.4 The Derivative Score: 0 of 3 pts 2.4.11 Find the indicated quantities for f(x) = 3x^2. (A) The slope of the secant line through the points (2,f(2)) and (2+h,f(2+ h)) as h approaches 0. (B) The slope of the graph at (2,f(2)). (C) The equation of the tangent line at (2,6). (A) The slope of the secant line through the points (2,f(2)) and (2+h,f(2+ h)) as h approaches 0, is Enter your answer in the answer box and then click Check Answer: 2 Pai remaining
The Correct Answer and Explanation is:
Correct Answer: 12 + 3h
Explanation
To find the slope of the secant line for the function f(x) = 3x², we use the slope formula, which is m = (y₂ – y₁) / (x₂ – x₁).
The problem gives us two points on the curve:
- Point 1: (x₁, y₁) = (2, f(2))
- Point 2: (x₂, y₂) = (2 + h, f(2 + h))
First, we need to calculate the y-coordinates for these points using the given function f(x) = 3x².
For Point 1:
y₁ = f(2) = 3(2)² = 3(4) = 12.
So, the first point is (2, 12).
For Point 2:
y₂ = f(2 + h) = 3(2 + h)².
To simplify this, we first expand the (2 + h)² term:
(2 + h)² = (2)² + 2(2)(h) + (h)² = 4 + 4h + h².
Now, substitute this back into the function:
y₂ = 3(4 + 4h + h²) = 12 + 12h + 3h².
Now we can plug these values into the slope formula for the secant line (also known as the difference quotient):
Slope (m_sec) = [f(2 + h) – f(2)] / [(2 + h) – 2]
Substitute the expressions we found:
m_sec = [(12 + 12h + 3h²) – 12] / [h]
Simplify the numerator:
m_sec = (12h + 3h²) / h
Now, factor out an ‘h’ from each term in the numerator:
m_sec = h(12 + 3h) / h
Since the problem states that h ≠ 0, we can safely cancel ‘h’ from the numerator and the denominator.
This leaves us with the final expression for the slope of the secant line:
m_sec = 12 + 3h
