Calculus Application Problems: Practice Here

Relative extrema of multivariables:
${\displaystyle f(x,y)=\frac{xy}{7}}$
find critical points and relative extrema, given an open region.

Rational exponents evaluate each expression.
(a) ${\displaystyle {27}^{1\text{/}3}}$
(b) ${\displaystyle {(-8)}^{1\text{/}3}}$
(c) ${\displaystyle -{\left(\frac{1}{8}\right)}^{1\text{/}3}}$

Solve the equation graphically in the given interval. State each answer rounded to two decimals.
${\displaystyle x^{\frac{1}{3}}-x=0,[-3,3]}$
$x=?$

Use the conjugate process ${\displaystyle \underset{x\to 0}{lim}\frac{\sqrt{x^2+5}-\sqrt{5-x^2}}{x}.}$

a) Find the Laplace transformation of the following function by using definition of Laplace transformation:
$f(t)=\cos (pt)\text{ where }t\ge 0$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The work required to move an object around a closed curve C in the presence of a vector force field is the circulation of the force field on the curve.
b. If a vector field has zero divergence throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is zero.
c. If the two-dimensional curl of a vector field is positive throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is positive (assuming counterclockwise orientation).

Higher-Order Derivatives
${\displaystyle f\left(x\right)=2x^4-3x^3+2x^2+x+4}$
find ${\displaystyle f^{10}\left(x\right)=?}$

Find both first partial derivatives. ${\displaystyle h(x,y)=e^{-(x2+y2)}}$

Find the differential dy for the given values of x and dx. $y=\frac{e^x}{10},x=0,dx=0.1$

An analysis of variance produces ${\displaystyle S{S}_{total}=40}$ and ${\displaystyle S{S}_{within}=10}$. For rhis analysis, what is ${\displaystyle S{S}_{between}}$?
a) 30
b) 400
c) Cannot be determined grom the information given
d) 50

The integrals converge. Evaluate the integrals without using tables
${\displaystyle {\int }_0^1\frac{4rdr}{\sqrt{1-r^4}}}$

Convert the above indefinite integrals into definate integrals using the intervals ${\displaystyle [0,1]}$.
(a)${\displaystyle \int \sqrt{a^2-x^2}dx}$
(b)${\displaystyle \int \sqrt{1-x^2}dx}$

$xydx-(x^2+3y^2)dy=0$

DIFFERENTIAL EQUATION
Given $f(t)=-\frac{1}{2t}+8,0\le t<4,f(t+4)=f(t)$
Find $F(s)=L\{f(t)\}$ of the Periodic Function

The laplace transform for ramp function with a time delay can be expressed as follows

Find the inverse laplace transform of
$\frac{6s+9}{s^2+17}s>0$
$y(t)=\dots$

Create a 2nd order, homogeneous, linear IVP, that is not guaranteed to have a unique solution at $x_0=3$

Solve. ${\displaystyle \int \frac{\sqrt{x}}{1+\sqrt[3]{x}}dx=?}$

Find the sum of the given vectors
${\displaystyle 50N\mathrm{\angle }{320}^{\circ }+170N\mathrm{\angle }{150}^{\circ }}$
$Rx=?$
$Ry=?$
$R=?$
$\theta =?$

Solve differential equation $dy/dx=e^{4}x(y-3)$