Master Calculus 1 Practice Problems with Expert Help

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) $z=g\left(xy\right)$

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?

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.

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.

Determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear.
${\displaystyle 5^2{u}_x+u_t=1+u_{\times }}$

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.
${\displaystyle f\left(x\right)=\frac{50}{e^{2x}}}$

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).

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.
${\displaystyle f(x,y)=\sqrt{x^2+y^2-4x+5}}$

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

Compute the gradient of the following functions and evaluate it at the given point P.
${\displaystyle g(x,y)=x^2-4x^{2}y-8x{y}^2;P(-1,2)}$

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

Determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear function that models the data. for those that are exponential, find an exponential function that models the data.
${\displaystyle xF\left(x\right)-1\left(\frac{1}{2}\right)0\left(\frac{1}{4}\right)1\left(\frac{1}{8}\right)2\left(\frac{1}{16}\right)3\frac{1}{32}}$

Determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear function that models the data. for those that are exponential, find an exponential function that models the data. xg(x) -12 05 18 211 314

James rents an apartment with an initial monthly rent of $1,600. He was told that the rent goes up 1.75% each year. Write an exponential function that models this situation to calculate the rent after 15 years. Round the monthly rent to the nearest dollar.

Evaluate the indefinite integral as a power series.
${\displaystyle \int \frac{{\tan }^{-1}x}{x}dx}$
${\displaystyle f\left(x\right)=C+\sum _{n=0}^{\mathrm{\infty }}\left(\dot{s}\right)}$
What is the radius of convergence R?

Find a suitable form for particular solution of the differential equation
${\displaystyle y\prime \prime +4y=e-2t(t^2+1)}$

${\int }_0^1\frac{1}{e^x+1}dx$