Section 2.8: The Derivative as a Function

 

1. Find the derivative of the function using the definition of derivative (do not use shortcuts).

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

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

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

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

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

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

 

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

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

graph

solution

 

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

 

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

 

6. The displacement (in meters) of a particle moving in a straight line is given by \( s(t) =t^3-2t\), where \(t\) is measured in seconds. Given that \(s'(t)=3t^2-2\), and \(s''(t)=6t\).   solution

(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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