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

Section 1-1 : The Shape of a Graph, Part I

The graph of a function is given below. Determine the intervals on which the function is concave up and concave down.
$The graph of a function on $ - 3 \le x \le 8$. Starting below the x-axis at the left the graph rises to a small peak at approximately $\left( { - 2,1} \right)$, decreases to the origin, then increases to a peak at approximately $\left( {4,5} \right)$. It then falls steeply, crossing the x-axis near $x = 6$, down to a valley at approximately $\left( {7, - 3} \right)$ and then turns back up to approximately $\left( {8, - 2} \right)$ at the right end.$
Solution
Below is the graph the 2^nd derivative of a function. From this graph determine the intervals in which the function is concave up and concave down.
$The graph of the second derivative of a function on $ - 5 \le x \le 4.5$. Coming down steeply from the upper left the graph crosses the x-axis at $x = - 4$, dips to a shallow valley just below the axis and crosses back up at $x = - 2$. It then rises to a peak at approximately $\left( {0,2} \right)$, decreases and touches the x-axis at $x = 3$ without crossing it, and then increases steeply to the upper right.$
Solution

For problems 3 – 8 answer each of the following.

$f\left( x \right) = 12 + 6{x^2} - {x^3}$ Solution
$g\left( z \right) = {z^4} - 12{z^3} + 84z + 4$ Solution
$h\left( t \right) = {t^4} + 12{t^3} + 6{t^2} - 36t + 2$ Solution
$h\left( w \right) = 8 - 5w + 2{w^2} - \cos \left( {3w} \right)$ on $\left[ { - 1,2} \right]$ Solution
$R\left( z \right) = z{\left( {z + 4} \right)^{\,{\frac{2}{3}}}}$ Solution
$h\left( x \right) = {{\bf{e}}^{4 - {x^{\,2}}}}$ Solution

For problems 9 – 14 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.
Determine the intervals on which the function is concave up and concave down.
Determine the inflection points of the function.
Use the information from steps (a) – (e) to sketch the graph of the function.

$g\left( t \right) = {t^5} - 5{t^4} + 8$ Solution
$f\left( x \right) = 5 - 8{x^3} - {x^4}$ Solution
$h\left( z \right) = {z^4} - 2{z^3} - 12{z^2}$ Solution
$Q\left( t \right) = 3t - 8\sin \left( {{\displaystyle \frac{t}{2}}} \right)$ on $\left[ { - 7,4} \right]$ Solution
$f\left( x \right) = {x^{\,{\frac{4}{3}}}}\left( {x - 2} \right)$ Solution
$P\left( w \right) = w{{\bf{e}}^{4w}}$ Solution
Determine the minimum degree of a polynomial that has exactly one inflection point. Solution
Suppose that we know that $f\left( x \right)$ is a polynomial with critical points $x = - 1$, $x = 2$ and $x = 6$. If we also know that the 2nd derivative is $f''\left( x \right) = - 3{x^2} + 14x - 4$. If possible, classify each of the critical points as relative minimums, relative maximums. If it is not possible to classify the critical points clearly explain why they cannot be classified. Solution