Calculus 2 Equations: Expert Solutions and More

Replace the Cartesian equation with equivalent polar equations. $xy=2$

Find $y^{\prime }(x)$ and $y^{″}(x)$ of the function $y=f(x)$ defined parametrically as ${\displaystyle x=t^2-3,y=t^3-2t,t=1}$

Polar coordinates:${\displaystyle (8,-\frac{\pi }{3})}$ convert into retangular coordinates

For the given rectangular equation, write an equivalent polar equation. $x^2+y^2-10x=0$

Express the point with polar coordinates ${\displaystyle p(2,-\frac{\pi }{4})}$ in rectangular coordinates

Writing a Vector-Valued function in exercise , represent the line segment from P to Q by a vector-valued function and by a set of parametric equations:
P(-2,5,-3),Q(-1,4,9)

Transform the second-order differential equation ${\displaystyle \frac{d^{2}x}{d{t}^2}=3x}$ into a system of first-order differential equations.

True or False? Justify your answer with a proof or a counterexample. You can determine the behavior of all first-order differential equations using directional fields or Euler's method.

Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m+mf. Continuing in this fashion, the amount of medication in your blood just after your nth does is ${\displaystyle A_n=m+mf+\dots +m{f}^{n-1}}$. For the given values off and m, calculate ${\displaystyle A_5,A_{10},A_{30}}$, and ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{A}_n}$. Interpret the meaning of the limit ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{A}_n}$. $f=0.25,$ $m=200mg.$

Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The function are real-valued. All rational functions.

Use the substitution $y^{\prime }=v$ to write each second-order equation as a system of two first-order differential equations (planar system). ${\displaystyle y^{{}^{″}}+\mu (t^2-1)y^{{}^{″}}+y=0}$

Suppose that a population develops according to the logistic equation
$\frac{dP}{dt}=0.05P-0.0005P^2$
where t is measured in weeks. What is the carrying capacity? What is the value of k?

If $x^2+xy+y^3=1$ find the value of y''' at the point where x = 1

Find $\frac{dy}{dx}$ using implicit differentiation $x{e}^y=x-y$

Consider the differential equation for a function f(t),
$tf"(t)+f^{\prime }(t)-f((t){)}^2=0$
a) What is the order of this differential equation?
b) Show that $f(t)=\frac{1}{t}$ is a particular solution to this differential equation.
c) Find a particular solution with $f(0)=0$
2. Find the particular solutions to the differential equations with initial conditions:
a)$\frac{dy}{dx}=\frac{\ln (x)}{y}$ with $y(1)=2$
b)$\frac{dy}{dx}=e^{4x-y}$ with $y(0)=0$

Transform the second-order differential equation ${\displaystyle \frac{d^{2}x}{d{t}^2}+\frac{dx}{dt}=x}$ into a system of first-order differential equations.

Find the exact length of the curve. Use a graph to determine the parameter interval.
$r={\cos }^2(\frac{\theta }{2})$

Transform the single linear differential equation into a system of first-order differential equations.
${\displaystyle x+tx+2t^{3}x-5t^4=0}$

Find an equation of the plane.
The plane through the points (2,1,2), (3,-8,6), and (-2,-3,1)

Let $P(x)=F(x)G(x)$ and $Q(x)=\frac{F(x)}{G(x)}$, where F and G are the functions whose graphs are shown.
a) Find $P^{\prime }(2)$
b) Find $Q^{\prime }(7)$