Show each step of your work and fully simplify each expression.
Turn in your answers in class on a physical piece of paper.
Staple multiple sheets together.
Feel free to use Desmos for graphing.
Answer the following:
Find the derivative of \[xy - y - 1 = 0\] in two ways:
Solve for $y$, then take the derivative normally.
Implicitly differentiate.
Show the two derivatives are identical (Hint: substitute the definition of $y$ in part (a)).
Find $\dfrac{dy}{dx}$ with implicit differentiation.
$x^2 + y^2 = 16$
$x^2 + 2y^3 + y^2 = x$
$x^4 + y^2 = y^4 + x^2$
$2 = x^2y^3 + y^2$
$(x+y)^3 + x^3 + y^3 = 0$
$\sqrt{xy} = x + y$
$x^2y^3 = 5 + 2xy^2$
$\dfrac{x^2}{y^3 + y^4} = y^2$
Find $\dfrac{d^2y}{dx^2}$ of the equation $xy= 1$.
Suppose a demand equation for cotton shirts is \[100x^2 + 9p^2 = 3600\] where $x$ represents (in thousands) the number of shirts demanded each week when the unit price is $\$p$. How fast is the quantity demanded increasing when the unit price per shirt is $\$14$ and the price is decreasing at $\$0.15$ shirt/week?
The formula for the volume $V$ of a cube with side length $s$ is $V = s^3$. The sides of the cube are 5 feet long and increasing at the rate of $0.2$ inches/second. How fast is the volume of the cube changing?
Suppose a drop of dye was dropped into a large bowl filled with water. If the dye spreads out in a circle and its radius is increasing at a rate of 2 cm/sec, determine how fast the area is increasing when the radius of the circle is 10 cm.
Suppose a meat distributor is willing to make $x$ pounds of beef available every week on the marketplace when the price is $\$p$ per pound. The relationship between quantity supplied and price is \[650p^2 - x^2 = 100\] If 30,000 pounds of beef are available on the marketplace this week and the price per pound is falling by 2 cents per week, at what rate is the supply falling?
Two people are standing next to each other. At the same time, one walks north at 4 mph, and the other walks east at 3 mph. How fast is the distance changing between them after thirty minutes? (Hint: be careful of units!)
The number of housing starts is related to the mortgage rate by the equation \[11N^{3}+r=16\] where $N(t)$ is the number of housing starts (in units of a million) and $r(t)$ is the mortgage rate (in percent per year). What is the rate of change of the mortgage rate with respect to time when the number of housing starts is 1 million and is decreasing by $1/33$ of a million per year?
Suppose \[9a^2 + 4b^2 = 36\] where $a$ and $b$ are functions of time. If $\dfrac{db}{dt} = \frac{1}{3}$, find $\dfrac{da}{dt}$ when $a = 2$ and $b = \dfrac{2}{3}\sqrt{5}$.