Behaviors of Implicit Relations (College Board AP® Calculus AB): Exam Questions

Consider the curve given by the equation $2-5xy=y^3$. It is known that $\frac{dy}{dx}=-\frac{5y}{3y^2+5x}$. Find the coordinates of a point on the curve at which the line tangent to the curve is horizontal, or explain why no such point exists.

Find the coordinates of a point on the curve at which the line tangent to the curve is vertical, or explain why no such point exists.

A particular type of medication is given to a patient. The amount, in milligrams, of the medication in the patient at time $t$ hours is modeled by a function $M$ that satisfies the differential equation $\frac{dM}{dt}=2-\frac{M}{3t+4}$. At time $t=1$ hour, there are $4$ milligrams of the medication in the patient. Is the rate of change of the amount of medication in the patient increasing or decreasing at $t=1$? Give a reason for your answer.

Consider the curve given by the equation $3y^3-6xy=6$. It can be shown that $\frac{dy}{dx}=\frac{2y}{3y^2-2x}$.

Find the coordinates of all points on the curve at which the line tangent to the curve at that point is vertical.

Evaluate $\frac{d^{2}y}{d{x}^2}$ at the point on the curve where $x=\frac{3}{2}$ and $y=2$.

Consider the curve with equation $\sin x+\sin y=1$.

The point $\left(\frac{\pi }{2},\pi \right)$ lies on the curve. Investigate whether this point is a critical point; and if it is, classify it as either a relative maximum point or a relative minimum point.

Find the coordinates of any points on the curve $y^2-x^2\sin \left(2y\right)=9$, where $\mathit{-}\pi \mathit{<}y\mathit{<}\pi$, at which the lines tangent to the curve are horizontal, or explain why no such points exist.

Consider the function $y=f\left(x\right)$ whose curve is given by the equation $18y^2-4=y\sin x$for $y>0$.

It is known that $\frac{dy}{dx}=\frac{y\cos x}{36y-\sin x}$.

For $0\le x\le \pi$ and $y>0$, find the coordinates of the point where the line tangent to the curve is horizontal.

Determine whether $f$ has a relative minimum, a relative maximum, or neither at the point found in part (a). Justify your answer.

Show that there are no horizontal or vertical tangent lines on the curve defined by $x^2+y^2+\ln \left(xy\right)=9$.

An enclosed water channel is being designed to maximise water flow in a garden irrigation system. It is to be constructed from a $100$ cm wide metal sheet. The channel's cross section consists of a rectangle above a semi-circle, sharing a straight edge. The semicircle has a radius of $r$ cm and the height of the rectangular part is $h$ cm.

The cross-sectional area is given by

$A=2hr+\frac{1}{2}\pi r^2$

The total perimeter of the cross-section (which includes the two vertical sides and the top length of the rectangle, as well as the semi-circular arc) must remain fixed at $100$ cm.

By first writing an equation for the total perimeter, find an expression for $\frac{dh}{dr}$ in terms of $h$ and $r.$ Then, find $\frac{dA}{dr}$ in terms of $h$ and $r.$

Hence, determine the value of $r$ that maximises $A.$ Justify your answer using a derivative test.

A large, conical ice sculpture is melting, causing its volume to decrease at a constant rate of $900$ cubic centimeters per minute. The ice sculpture takes the form of a cone, with both its radius and height changing with time. (Note: The volume $V$ of a cone with radius $r$ and height $h$ is given by $V=\frac{1}{3}\pi r^{2}h$.)

Show that $\frac{dV}{dt}=\frac{1}{3}\pi r\left(r\frac{dh}{dt}+2h\frac{dr}{dt}\right)$, then find an expression for $\frac{dh}{dt}$ in terms of $\frac{dr}{dt}$ at the instant when the radius of the sculpture is 60 centimeters and the height is 120 centimeters.

Given that the height of the cone is always decreasing, analyze the behavior of the rate of change of the radius, $\frac{dr}{dt}$, as $\frac{dh}{dt}$ changes.

Then find the minimum value of $\frac{dh}{dt}$ needed for the radius of the cone to be shrinking.

A drainage system is put into place to remove the melted ice. The rate at which the water is removed is given by $D\left(t\right)=300\sqrt{t}$ cubic centimeters per minute, where $t$ is the time in minutes since the drainage system began working. The ice continues to melt at a rate of 900 cubic centimeters per minute. Find the time, $t$ at which the puddle of water reaches its maximum volume. Justify your answer.

Find the coordinates of any points on the curve $3x^2\cos \left(3y\right)=2y^3-16$, where $0\mathit{<}y\mathit{<}\frac{\pi }{2}$, at which the lines tangent to the curve are horizontal. Then, determine whether each point has a relative minimum, a relative maximum, or neither. Justify your answer.

It is known that $\frac{dy}{dx}=\frac{6x\cos \left(3y\right)}{6y^2+9x^2\sin \left(3y\right)}$.