# Homework 6

Directions:

1. Show each step of your work and fully simplify each expression.
2. Turn in your answers in class on a physical piece of paper.
3. Staple multiple sheets together.
4. Feel free to use Desmos for graphing.

1. You stood on the edge of a building 225 feet tall and dropped a ball. The distance (in feet) the ball falls in $t$ seconds is $f(t) = 15t^2$
1. At what time does the ball hit the ground?
2. What is the ball's velocity when it hits the ground?
3. What is the ball's acceleration when it hits the ground?
2. The Consumer Price Index for some economy is predicted to follow the function $f(t) = -0.0003t^{3}+0.009t^{2}+0.0055t+1.29$ where $f(t)$ is in dollars in tens of thousands and $t = 0$ is the beginning of year 2023.
1. What is the inflation rate in the beginning of 2040?
2. Show the rate at which inflation is growing is slowing down in 2040.
3. Show the rate at which inflation is growing is speeding up in 2028.
3. Obesity (measured by BMI) in America is approximated by the function $f(t) = 0.0004t^3 + 0.0036t^2 + 0.8t + 12 \qquad 0 \leq t \leq 13$ where $t = 0$ (in years) is the beginning of 1991 and $f(t)$ is percent of people who are obese.
1. Show $f(t)$ is an increasing function on $0 \leq t \leq 13$. (Hint: $t$ is positive, is each term in $f'(t)$ positive?)
2. What are the implications of $f(t)$ being an increasing function?
3. Show the rate of change of the rate of change of the percent of people deemed obese was positive from 1991 to 2004. What are the implications?
4. Find the derivative of $xy - y - 1 = 0$ in two ways:
1. Solve for $y$, then take the derivative normally.
2. Implicitly differentiate.
3. Show the two derivatives are identical (Hint: substitute the definition of $y$ in part (a)).
5. Find $\dfrac{dy}{dx}$ with implicit differentiation.
1. $x^2 + y^2 = 16$
2. $x^2 + 2y^3 + y^2 = x$
3. $x^4 + y^2 = y^4 + x^2$
4. $2 = x^2y^3 + y^2$
5. $(x+y)^3 + x^3 + y^3 = 0$
6. $\sqrt{xy} = x + y$
7. $x^2y^3 = 5 + 2xy^2$
8. $\dfrac{x^2}{y^3 + y^4} = y^2$
6. Find $\dfrac{d^2y}{dx^2}$ of the equation $xy= 1$.