Calculus I - The Shape of a Graph, Part I (Practice Problems)

Section 4.5 : The Shape of a Graph, Part I

For problems 1 & 2 the graph of a function is given. Determine the intervals on which the function increases and decreases.

$The graph of a function on $ - 5 \le x \le 9$. Starting at approximately $\left( { - 5,4} \right)$ the graph decreases to a valley at approximately $\left( { - 3, - 1} \right)$, then increases to a peak at approximately $\left( {1,2} \right)$, then decreases the whole way down to a valley at approximately $\left( {7, - 5} \right)$ and finally increases to approximately $\left( {9, - 2} \right)$ at the right end.$
Solution
$The graph of a function on $0 \le x \le 9.5$. The graph rises steeply from below the x-axis, crosses the x-axis at the origin and continues up to a peak at approximately $\left( {1,4} \right)$. It then decreases to a valley at approximately $\left( {4, - 2} \right)$, then increases again, flattening out into a short nearly level stretch at about $y = 2$ between $x = 7$ and $x = 8.5$, and then rises to approximately $\left( {9.5,3} \right)$ at the right end.$
Solution

Below is the graph of the derivative of a function. From this graph determine the intervals in which the function increases and decreases.

$The graph of the derivative of a function on $ - 8 \le x \le 5$. Starting well below the x-axis at the left the graph rises steeply, crosses the x-axis near $x = - 7$, reaches a peak at approximately $\left( { - 6,1.5} \right)$, then decreases and just touches the x-axis at $x = - 2$ without crossing it. From there it increases to a peak at approximately $\left( {3,3.5} \right)$ and then falls steeply, crossing the x-axis at $x = 5$ and continuing down off the bottom of the graph.$
Solution
This problem is about some function. All we know about the function is that it exists everywhere and we also know the information given below about the derivative of the function. Answer each of the following questions about this function.
1. Identify the critical points of the function.
2. Determine the intervals on which the function increases and decreases.
3. Classify the critical points as relative maximums, relative minimums or neither.
\[\begin{array}{c}f'\left( { - 5} \right) = 0\,\,\,\,\,\,\,\,f'\left( { - 2} \right) = 0\,\,\,\,\,\,\,\,\,\,f'\left( 4 \right) = 0\,\,\,\,\,\,\,\,\,\,\,f'\left( 8 \right) = 0\\ f'\left( x \right) < 0\,\,\,\,\,{\rm{on}}\,\,\,\,\,\left( { - 5, - 2} \right),\,\,\,\left( { - 2,4} \right),\,\,\,\left( {8,\infty } \right)\hspace{0.25in}\hspace{0.25in}f'\left( x \right) > 0\,\,\,\,\,{\rm{on}}\,\,\,\,\,\left( { - \infty , - 5} \right),\,\,\,\left( {4,8} \right)\end{array}\] Solution

For problems 5 – 12 answer each of the following.

Identify the critical points of the function.
Determine the intervals on which the function increases and decreases.
Classify the critical points as relative maximums, relative minimums or neither.

$f\left( x \right) = 2{x^3} - 9{x^2} - 60x$ Solution
$h\left( t \right) = 50 + 40{t^3} - 5{t^4} - 4{t^5}$ Solution
$y = 2{x^3} - 10{x^2} + 12x - 12$ Solution
$p\left( x \right) = \cos \left( {3x} \right) + 2x$ on $\displaystyle \left[ - \frac{3}{2},\,\,2 \right]$ Solution
$\displaystyle R\left( z \right) = 2 - 5z - 14\sin \left( {\frac{z}{2}} \right)$ on $\left[ { - 10,7} \right]$ Solution
$h\left( t \right) = {t^2}\,\sqrt[3]{{t - 7}}$ Solution
$f\left( w \right) = w{{\bf{e}}^{2\, - \,\,{\frac{1}{2}}{w^{\,2}}}}$ Solution
$g\left( x \right) = x - 2\ln \left( {1 + {x^2}} \right)$ Solution
For some function, $f\left( x \right)$, it is known that there is a relative maximum at $x = 4$. Answer each of the following questions about this function.
1. What is the simplest form for the derivative of this function?
  Note : There really are many possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative that you can come up with.
2. Using your answer from (a) determine the most general form of the function.
3. Given that $f\left( 4 \right) = 1$ find a function that will have a relative maximum at $x = 4$.
  Note : You should be able to use your answer from (b) to determine an answer to this part.
Solution
Given that $f\left( x \right)$ and $g\left( x \right)$ are increasing functions. If we define $h\left( x \right) = f\left( x \right) + g\left( x \right)$ show that $h\left( x \right)$ is an increasing function. Solution
Given that $f\left( x \right)$ is an increasing function and define $h\left( x \right) = {\left[ {f\left( x \right)} \right]^2}$. Will $h\left( x \right)$ be an increasing function? If yes, prove that $h\left( x \right)$ is an increasing function. If not, can you determine any other conditions needed on the function $f\left( x \right)$ that will guarantee that $h\left( x \right)$ will also increase? Solution