Finding a Tangent Line to a Curve
Use implicit differentiation to find $y^{\prime}$, and then evaluate $y^{\prime}$ for $x^{2} y-3 x^{2}-4=0$ at the point $(2,4)$.
\[
\begin{array}{l}
y^{\prime}=\square \\
\left.\left.y^{\prime}\right|_{(2,4)}=\square \text { (Simplify your answer. }\right)
\end{array}
\]
Checking if Differentiable Over an Interval
Check if the function \(f(x) = |x|\) is differentiable over the interval \([-2,2]\).
The Mean Value Theorem
Find the average rate of change of $g(x)=x^{3}-3 x^{2}+2 x$ from $x=-2$ to $x=1$. Simplify your answer as much as possible.
Finding the Inflection Points
8. Find the value of the constant $b$ such that the following function has a point of inflection at $x=3$. $f(x)=\sqrt{x+1}+\frac{b}{x}$
Find Where the Function Increases/Decreases
Part 2 of 5
For the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing or decreasing; $c$ ) give the coordinates of any points of inflection; d) identify intervals where the function is concave up or concave down, and e) sketch the graph.
\[
h(x)=3 x^{3}-9 x
\]
b) On what interval(s) is $\mathrm{h}$ increasing or decreasing? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The function is increasing on $\square$. The function is decreasing on (Simplify your answers. Type your answers in interval notation. Use a comma to separate answers as needed.)
B. The function is increasing on The function is never decreasing.
(Simnlify your answer. Type your answer in interval notation. Use a comma to separate answers
Finding the Critical Points of a Function
Determine all critical points for the following function.
\[
f(x)=x^{2}-256 \sqrt{x}
\]
Find Horizontal Tangent Line
For the function, find the point(s) on the graph at which the tangent line has slope 5
\[
y=\frac{1}{3} x^{3}-4 x^{2}+20 x+9
\]
The point(s) is/are
(Simplify your answer. Type an ordered pair. Use a comma to separate answers as needed.)
Evaluating Limits with L'Hospital Rule
Evaluate the limit \(\lim_{{x \to 0}} \frac{{e^x - 1}}{{x}}\)
Local Maxima and Minima
A rectangular page is to contain 27 square inches of print. The page has to have a 3-inch margin on top and at the bottom and a 4-inch margin on each side (see figure). Find the dimensions of the page that minimize the amount of paper used.
The dimensions that minimize the amount of paper used are in.
(Simplify your answers. Use a comma to separate answers.)
Finding the Absolute Maximum and Minimum on the Given Interval
A rectangular garden of area 300 square feet is to be surrounded on three sides by a brick wall costing $\$ 10$ per foot and on one side by a fence costing $\$ 5$ per foot. That is, two sides of equal length will consist of brick walls, and the other two sides of equal length will consist of one brick wall and a fence. Find the dimensions of the garden such that the cost of the materials is minimized.
To minimize costs, the lengths of the sides consisting of a fence and a brick wall should be $\square$ feet and the lengths of the perpendicular sides, which are only made from brick walls, should be feet.
Finding Concavity using the Second Derivative
Let's consider the function \(f(x) = 2x^3 - 9x^2 + 12x - 3\). Find the intervals of concavity and the inflection points.
Using the Limit Definition to Find the Tangent Line at a Given Point
Use $\Delta y \approx f^{\prime}(x) \Delta x$ to find a decimal approximation of the radical expression.
\[
\sqrt[3]{8.66}
\]
What is the value found by using $\Delta y \approx f^{\prime}(x) \Delta x$ ?
\[
\sqrt[3]{8.66} \approx \square \text { (Round to three decimal places as needed.) }
\]
Finding the Derivative using the Fundamental Theorem of Calculus
The surface area of a human (in square meters) has been approximated by A $=0.024265 \mathrm{~h}^{0.3964} \mathrm{~m}^{0.5378}$, where $\mathrm{h}$ is the height (in $\mathrm{cm}$ ) and $\mathrm{m}$ is the mass (in $\mathrm{kg}$ )
(a) Find the approximate change in surface area if the mass changes from $69 \mathrm{~kg}$ to $70 \mathrm{~kg}$, while the height remains $184 \mathrm{~cm}$. Use the derivative to estimate the change
(b) Find the approximate change in surface area when the height changes from $165 \mathrm{~cm}$ to $166 \mathrm{~cm}$, while the mass remains at $75 \mathrm{~kg}$. Use the derivative to estimate the change
(a) Differentiate $A(m, h)$ with respect to $m$ to find $A_{m}(m, h)$
$A_{m}(m, h)=\frac{\partial}{\partial m}\left(0.024265 h^{0.3964} m^{0.5378}\right)=$
(Round to five decimal places as needed )
Find the Turning Points
Find the turning points of the function \(f(x) = x^3 - 3x^2 - 9x + 5\).