Derivative as a Function

1. Find the derivative using the limit definition:

Problem 1(a)

\( \displaystyle{f(x)= x^2+2x} \)

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Problem 1(b)

\( \displaystyle{f(x)= 1- x^2} \)

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Problem 1(c)

\( \displaystyle{f(x)= x^2-5x+1}\)

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Problem 1(d)

\( \displaystyle{f(x)= 3x^2-5x }\)

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Problem 1(e)

\( \displaystyle{f(x)= \frac{1}{x}} \)

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Problem 1(f)

\( \displaystyle{f(x)= \frac{1}{\sqrt{x}}} \)

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2. Applications and Concepts:

Problem 2

Use the definition of a derivative to find \(f'\) and \(f''.\) Given that \(f(x)=3x^2-5x+7.\)

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Problem 3

The graph of \( y = f(x) \) is given below. State, with reasons, the numbers at which \( f \) is not differentiable.

Graph of a function with corners, discontinuities, or vertical tangents.

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Problem 4

Find an equation of the tangent line to the curve \( f(x) =\sqrt{x} \) at \(x=4\). Given that \( f'(x)=\dfrac{1}{2\sqrt{x}} \).

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Problem 5

Find an equation of the tangent line to the curve \( f(x) = 2x^2-5x \) at \( (1, -3)\). Given that \( f'(x)= 4x-5 \).

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Problem 6

A particle moves in a straight line with displacement \( s(t) =t^3-2t\). Given \(s'(t)=3t^2-2\) and \(s''(t)=6t\):

(a) Find the average velocity over the time interval \( [2, 5]\).
(b) Find the instantaneous velocity at \(t=2\).
(c) Find the acceleration when \( t = 3\).

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