In this exercise, the logistic equation models the growth of a population. Use the equation to determine when the population will reach $50 \%$ of its carrying capacity.

$$P(t)=\dfrac{1500}{1+24e^{-0.75t}}$$

In this exercise, the logistic equation models the growth of a population. Use the equation to find the carrying capacity.

$$P(t)=\dfrac{1500}{1+24e^{-0.75t}}$$

In this exercise, the logistic equation models the growth of a population. Use the equation to find the value of $k$.

$$P(t)=\dfrac{1500}{1+24e^{-0.75t}}$$

In hilly areas, radio reception may be poor. Consider a situation where an FM transmitter is located at the point $(-1,1)$ behind a hill modeled by the graph of

$$y=x-x^2$$

and a radio receiver is located on the opposite side of the hill. (Assume that the $x$-axis represents ground level at the base of the hill.)

\What is the closest position $(x, 0)$ the radio can be to the hill so that reception is unobstructed?

A not uncommon calculus mistake is to believe that the product rule for derivatives says that $(f g)^{\prime}=f^{\prime} g^{\prime}$. If $f(x)=e^{x^2}$, determine, with proof, whether there exists an open interval $(a, b)$ and a nonzero function $g$ defined on $(a, b)$ such that this wrong product rule is true for $x$ in $(a, b)$.

In this exercise, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

The families

$$x^2+y^2=2Cy\text{ and }x^2+y^2=2Kx$$

are mutually orthogonal.

In this exercise, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

The function

$$f(x, y)=x^2+x y+2$$

is homogeneous.

In this exercise, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

The differential equation

$$y^{\prime}=x y-2 y+x-2$$

can be written in separated variables form.

Prove that if the family of integral curves of the differential equation

$$\dfrac{dy}{dx}+p(x)y=q(x),\qquad p(x)\cdot q(x)\neq0$$

is cut by the line $x=k$ the tangents at the points of intersection are concurrent.

In this exercise, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

The function $y=0$ is always a solution of a differential equation that can be solved by separation of variables.

Ignoring resistance, a sailboat starting from rest accelerates $(dv/dt)$ at a rate proportional to the difference between the velocities of the wind and the boat.

Use the result of previous part to write the distance traveled by the boat as a function of time.

Ignoring resistance, a sailboat starting from rest accelerates $(dv/dt)$ at a rate proportional to the difference between the velocities of the wind and the boat.

The wind is blowing at $20$ knots, and after $1$ minute the boat is moving at $5$ knots. Write the velocity $v$ as a function of time $t$.

Let $f$ be a twice-differentiable real-valued function satisfying

$$f(x)+f^{\prime\prime}(x)=-xg(x)f^{\prime\prime}(x)$$

where $g(x)\geq0$ for all real $x$. Prove that $|f(x)|$ is bounded.

It is known that $y=A\sin\omega t$ is a solution of the differential equation $y^{\prime\prime}+16y=0$. Find the values of $\omega$.