7.1.5How do I measure a spinning object?

STEEPNESS IN SEATTLE, Part One

Seattle’s Space Needle officially opened on the first day of the Seattle World’s Fair, April 21, 1962. It features a flying saucer-like revolving restaurant $500$ feet above the ground. The restaurant is accessible by stairs ($832$ steps), but most people take an elevator.

Suppose you are across the street having coffee, watching the elevator ascend and descend. Your window is $100$ feet above the ground and $2000$ feet away from the Space Needle. The elevator ascends and descends at a constant speed of $14$ ft/sec. Let $θ$ be the angle of elevation of your line of sight to the elevator and $h\left(t\right)$ is the height of the elevator above ground.

1. Make a prediction: Describe how $\frac { d \theta } { d t }$ changes. Explain what your answer means physically (as you watch the elevator rise). 

2. Write an equation for $\frac { d \theta } { d t }$.

3. Interpret the equation for $\frac { d \theta } { d t }$ in the context of this problem. What does $\frac { d \theta } { d t }$ represent physically?

4. At what height will the elevator be when it appears to be moving fastest? What is $θ$ at this height? Use your equation for $\frac { d \theta } { d t }$ to support your answer.

Differentiate each of the following functions. Homework Help ✎

1. $y=2^{3x^2−x}+6x^{−0.4}$ 

2. $y=6x\ln(−3y)−x^{6^2}$ 

3. $y = \frac{3}{7}\cos(11^{0.2x})$ 

Integrate. Homework Help ✎

As Khalid inflates a spherical balloon, Kareem wonders about its different rates. He knows that the rate at which Khalid blows is equal to the rate at which the volume changes $( \frac { d V } { d t } )$. As the balloon inflates, other aspects are changing as well, such as the radius and the surface area. Homework Help ✎

1. If  $\frac { d V } { d t }=10 \ \frac { \text{cm} ^ { 3 } } { \text{sec} }$, calculate the rate of change of the radius, $\frac { d r } { d t }$, when $r = 3$ cm.

2. If $\frac { d V } { d t }= 12 \ \frac { \text{cm} ^ { 3 } } { \text{sec} }$, calculate the rate of change of the surface area, $\frac { d A } { d t }$, when $r = 5$ cm.

3. Describe is happening to the balloon when $\frac { d V } { d t }$ is negative.

Use the first and second derivatives to determine the following locations for $f(x) = xe^x$. Homework Help ✎

1. Relative minima and maxima

2. Intervals over which $f$ is increasing and decreasing

3. Inflection points

4. Intervals over which $f$ is concave up and concave down

The point from problem 7-19 travels along the $x$-axis so that at time $t$ its position is given by $s(t) = t^3 - 5t^2 + 4t$, where $0 ≤ t ≤ 5$. Calculate the average velocity of the point. At what time(s) during the interval was the point traveling at this average velocity? Homework Help ✎

Let $f^\prime(x) =\frac { \operatorname { ln } | x - 2 | } { 3 }$ and $f(−3) =\frac { 7 } { 3 }$. Homework Help ✎

1. Estimate $f(−3.1)$. 

2. Find $f^{\prime\prime}(x)$? 

3. Use concavity to determine if your answer to part (a) is an underestimate or an overestimate of the actual value of $f(–3.1)$. Justify your answer.