91Ó°ÊÓ

Prove Theorem 6.20 by solving the initial-value problem $\frac{dQ}{dt}=kQwithQ\left(0\right)=Q_0$, where k is a constant

Prove Theorem 6.21 by solving the initial-value problem $\frac{dP}{dt}=rP1-\frac{P}{K}$with P(0) = P0, where r and K are constants

Use Definition 6.6, the Mean Value Theorem ,and the definition of the definite integral to prove Theorem 6.7: The arc length of a sufficiently well behaved function f(x) on an interval [a, b] can be represented by the definite$I={∫}_a^b\sqrt{1+{\left(f\text{'}\left(x\right)\right)}^{2}dx}$

Prove, in two ways, that the arc length of a linear function $f\left(x\right)=mx+c$on an interval $\left[a,b\right]$is equal to $(b-a)\sqrt{1+m^2}$: (a) by using the distance formula; (b) by using Theorem 6.7.

Use the solution of the logistic model

$\frac{dP}{dt}=rP1-\frac{P}{K}$

to prove that as t →∞, the population P(t) approaches

the carrying capacity L. Assume that the constant k is positive.

Prove Theorem 6.22 by solving the initial-value problem $\frac{dT}{dt}=k(A−T)$with T(0) = T0, where k and A are constants.

Use Theorem 6.7 to prove that a circle of radius 5 has circumference$10\mathrm{Ï€}$.

Use the solution of the differential equation $\frac{dT}{dt}=k(A−T)$for the Newton’s Law of Cooling and Heating model to prove that as t → ∞, the temperature T(t) of an object approaches the ambient temperature A of its environment. The proof requires that we assume that k is positive. Why does this make sense regardless of whether the model represents heating or cooling?

Use Theorem 6.7 to prove that a circle of radius $r$has circumference$2\mathrm{Ï€°ù}$.

Explain, using the chain rule and / or $u$-substitution, why

$∫\frac{1}{q\left(y\right)}\frac{dy}{dx}dx=∫\frac{1}{q\left(y\right)}dy$.