Discover and Master Analyzing functions with Our Comprehensive Resources and Practice Problems

Find the critical points of the following functions.
${\displaystyle f(x,y)=y{e}^x-e^y}$

Without integrating, explain why
${\displaystyle {\int }_{-2}^{2}x{(x^2+1)}^{2}dx=0}$

Analyzing the Motion of a Projectile A projectile is fired
at an inclination of ${\displaystyle {45}^{\circ }}$ to the horizontal, with a muzzle
velocity of 100 feet per second. The height h of the projectile
is modeled by ${\displaystyle h\left(x\right)=\frac{-32x^2}{{100}^2}+x}$
where x is the horizontal distance of the projectile from the firing point
(a) At what horizontal distance from the firing point is the height of the projectile a maximum?
(b) Find the maximum height of the projectile.
(c) At what horizontal distance from the firing point will the projectile strike the ground?
(d) Using a graphing utility, graph the function h, ${\displaystyle 0\le x\le 350}$ .

Analyzing critical points Use the Second Derivative Test to classify the critical points of ${\displaystyle f(x,y)=xy(x-2)(y+3)}$.

List the various rules that allow us to differentiate constant functions,
power functions, polynomials, exponential functions, rational functions, and certain combinations of them,
simply and directly, without having to take limits each time?

Plot the graphs of the given functions
$f(x)=\{\begin{array}{ll}|x|& forx<2\\ -x+1& forx\ge 2\end{array}$
$m(x)=\{\begin{array}{ll}3& for-4<x<-1\\ -x& for-1\le x<3\\ \sqrt{x-3}& forx\ge 3\end{array}$

What are the derivatives of the inverse trigonometric functions? How do the domains of the derivatives compare with the domains of the functions?
The inverse trigonometric functions are
${\displaystyle {\sin }^{-1}x}$, ${\displaystyle {\cos }^{-1}x}$, ${\displaystyle {\tan }^{-1}x}$, ${\displaystyle {\csc }^{-1}x}$, ${\displaystyle {\sec }^{-1}x}$, ${\displaystyle {\cot }^{-1}x}$

Suppose f(t) and g(t) are single variable differentiable functions.Find ${\displaystyle \frac{\partial z}{\partial x}}$ and ${\displaystyle \frac{\partial z}{\partial y}}$ for each of the following binary functions.
(a) $z=f\left(x\right)g\left(y\right)$
(b) $z=g\left(xy\right)$

Consider ${\displaystyle f\left(x\right)=x{e}^{-kx}}$ for ${\displaystyle k>0}$. Find the relative extrema and the points of inflection of the function.

Suppose you are working for a manufacturing company that is analyzing a newer product using the following:
${\displaystyle A=\pi r^2}$
${\displaystyle \frac{dA}{dt}=2\pi r\left\{\left(\frac{dr}{dt}\right\}\right)}$ Explain what the product engineers may be examining/testing/analyzing. Why is the factor ${\displaystyle \left\{dr\right\}\left\{dt\right\}}$ needed?

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.
${\displaystyle f\left(x\right)=\frac{50}{e^{2x}}}$

Techniques of Integration: Integration by Parts, Products of Powers of Trigonometric Functions
Use integration by parts to integrate functions.
Integrate products of powers of trigonometric functions.
Evaluate
${\displaystyle \int {\csc }^{3}x{\cot }^{5}xdx}$

Calculus studies change by analyzing slopes of secant lines over successively shorter intervals.Determine whether the statement makes sense or does not make sense, and explain your reasoning.

A different interpretation of marginal cost Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and tak- ing into account storage costs, it has been determined that the total cost ${\displaystyle C\left(x\right)}$ of producing 25,000 gadgets in batches of x items at a time is given by
${\displaystyle C\left(x\right)=1250000+\frac{125000000}{x}+1.5x}$
a. Determine the marginal cost and average cost functions. Graph and interpret these functions.
b. Determine the average cost and marginal cost when ${\displaystyle x=5000}$.
c. The meaning of average cost and marginal cost here is differ- ent from earlier examples and exercises. Interpret the meaning of your answer in part (b).

Evaluate the following:
1. ${\displaystyle F={\int }_1^e{\ln }^{2}tdt}$

Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points.
${\displaystyle f(x,y)=\sqrt{x^2+y^2-4x+5}}$

Assuming the given information is accurate, determine whether the following statements are true. Provide an explanation.
Sixty percent of the students at an orientation are men and ${\displaystyle 30\mathrm{\%}}$ of the students at the orientation are arts majors. Therefore, ${\displaystyle 60\mathrm{\%}\times 30\mathrm{\%}=18\mathrm{\%}}$ of the students at the orientation are male arts majors.

${\displaystyle \sin \left\{(t+2\pi \}\right)}$=? and ${\displaystyle \cos \left\{(t+2\pi \}\right)}$=?, so the sine and cosine functions are ? function. The period of each of these is ?.

What can be said about the sum and difference of each pair of functions?
(a) Two even functions
(b) Two odd functions
(c) An odd function and an even function

Let $g(x)={\int }_0^{x}f(t)dt$ where f is the function whose graph is shown in the figure.
(a) Estimate g(0), g(2), g(4), g(6), and g(8).
(b) Find the largest open interval on which g is increasing. Find the largest open interval on which g is decreasing.
(c) Identify any extrema of g.
(d) Sketch a rough graph of g.