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Section 32 : Interpretation of the Derivative
For problems 1 – 3 use the graph of the function, \(f\left( x \right)\), estimate the value of \(f'\left( a \right)\) for the given values of \(a\).

 \(a =  5\)
 \(a = 1\)

 \(a =  2\)
 \(a = 3\)

 \(a =  3\)
 \(a = 4\)
For problems 4 – 6 sketch the graph of a function that satisfies the given conditions.
 \(f\left( {  7} \right) = 5\), \(f'\left( {  7} \right) =  3\), \(f\left( 4 \right) =  1\), \(f'\left( 4 \right) = 1\)
 \(f\left( 1 \right) = 2\), \(f'\left( 1 \right) = 4\), \(f\left( 6 \right) = 2\), \(f'\left( 6 \right) = 3\)
 \(f\left( {  1} \right) =  9\), \(f'\left( {  1} \right) = 0\), \(f\left( 2 \right) =  1\), \(f'\left( 2 \right) = 3\), \(f\left( 5 \right) = 4\), \(f'\left( 5 \right) =  1\)
For problems 7 – 9 the graph of a function, \(f\left( x \right)\), is given. Use this to sketch the graph of the derivative, \(f'\left( x \right)\).
 Answer the following questions about the function \(g\left( z \right) = 1 + 10z  7{z^2}\).
 Is the function increasing or decreasing at \(z = 0\)?
 Is the function increasing or decreasing at \(z = 2\)?
 Does the function ever stop changing? If yes, at what value(s) of \(z\) does the function stop changing?
 What is the equation of the tangent line to \(f\left( x \right) = 5x  {x^3}\) at \(x = 1\).
 The position of an object at any time \(t\) is given by \(s\left( t \right) = 2{t^2}  8t + 10\).
 Determine the velocity of the object at any time \(t\).
 Is the object moving to the right or left at \(t = 1\)?
 Is the object moving to the right or left at \(t = 4\)?
 Does the object ever stop moving? If so, at what time(s) does the object stop moving?
 Does the function \(R\left( w \right) = {w^2}  8w + 20\) ever stop changing? If yes, at what value(s) of \(w\) does the function stop changing?
 Suppose that the volume of air in a balloon for \(0 \le t \le 6\)is given by\(V\left( t \right) = 6t  {t^2}\) .
 Is the volume of air increasing or decreasing at \(t = 2\)?
 Is the volume of air increasing or decreasing at \(t = 5\)?
 Does the volume of air ever stop changing? If yes, at what times(s) does the volume stop changing?
 What is the equation of the tangent line to \(f\left( x \right) = 5x + 7\) at \(x =  4\)?
 Answer the following questions about the function \(Z\left( x \right) = 2{x^3}  {x^2}  x\).
 Is the function increasing or decreasing at \(x =  1\)?
 Is the function increasing or decreasing at \(x = 2\)?
 Does the function ever stop changing? If yes, at what value(s) of \(x\) does the function stop changing?
 Determine if the function\(V\left( t \right) = \sqrt {14 + 3t} \) increasing or decreasing at the given points.
 \(t = 0\)
 \(t = 5\)
 \(t = 100\)
 Suppose that the volume of water in a tank for \(t \ge 0\) is given by \(\displaystyle Q\left( t \right) = \frac{{{t^2}}}{{t + 2}}\).
 Is the volume of water increasing or decreasing at \(t = 0\)?
 Is the volume of water increasing or decreasing at \(t = 3\)?
 Does the volume of water ever stop changing? If so, at what times(s) does the volume stop changing?
 What is the equation of the tangent line to \(g\left( x \right) = 10\) at \(x = 16\)?
 The position of an object at any time \(t\) is given by \(Q\left( t \right) = \sqrt {1 + 4t} \).
 Determine the velocity of the object at any time \(t\).
 Does the object ever stop moving? If so, at what time(s) does the object stop moving?
 Does the function \(Y\left( t \right) = 2{t^3} + 9t + 5\) ever stop changing? If yes, at what value(s) of \(t\) does the function stop changing?