Questions

MAT135H5_F25_ALL SECTIONS 3.1 Preparation Check

Multiple choice

Which of the following are equivalent to asking "Find the derivative of 𝑓 ( 𝑥 ) = 𝑥 2 at 𝑥 = 3 "? (Select all that apply.)

Options

A.Find 𝑓 ′ ( 3 ) if 𝑓 ( 𝑥 ) = 𝑥 2 .

B.Compute the limit lim ℎ ⟶ 0 ( 3 + ℎ ) 2 − 3 2 ℎ

C.Find the average velocity of 𝑓 ( 𝑥 ) = 𝑥 2 on the interval [ 0 , 3 ] .

D.Compute the limit lim 𝑥 ⟶ 3 ( 3 + 𝑥 ) 2 − 3 2 𝑥 − 3

E.Find the difference quotient of 𝑓 ( 𝑥 ) = 𝑥 2 at 𝑥 = 3 .

F.Find the slope of the tangent line to the graph of 𝑓 ( 𝑥 ) = 𝑥 2 at the point ( 3 , 9 ) .

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Step-by-Step Analysis

We’re evaluating which statements are equivalent to finding the derivative of f(x) = x^2 at x = 3. Option 1: Find the difference quotient of f(x) = x^2 at x = 3. This is a direct restatement of the derivative definition: [f(3+h) − f(3)]/h as h → 0. Since f(3) = 9 and f(3+h) = (3+h)^2, this expression becomes [(3+h)^2 − 9]/h, which indeed yields f′(3). Therefore, this optio......Login to view full explanation

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