11.3.2Anyone want to hit the slopes?

More Slopes of Polar Curves

11-90.

Use the method of your choice to write an expression for the slope of each of the following polar curves.

$r = 5\cos(θ)$

$r =\frac { 2 } { \sqrt { \operatorname { cos } ( \theta ) \cdot \operatorname { sin } ( \theta ) } }$

$r = 2 + 3\cos(θ)$

$r^2 = 4\sin(2θ)$

11-91.

The curves $r = 1 + \cos(θ)$ and $r = 5(1 + \cos(θ))$ are cardioids of different sizes.

How does the slope, $\frac { d y } { d x }$ , of the curve $r = 1 + \cos(θ)$ compare to the slope of the curve $r = 5(1 + \cos(θ))$ for any angle $θ$ ?
What is the slope, $\frac { d y } { d x }$ , of the curve $r = a(1 + \cos(θ))$ where $a$ is a constant?
In general, what conclusion can you make about the slopes of two curves $r = 1 + \cos(θ) \text{ and }r = a(1 + \cos(θ))$ , where $a$ is a constant?

11-92.

Consider $r = 1 + \cos(θ)$ and $r = 5(1 + \cos(θ))$ from problem 11-91. Calculate the area of the region bounded by each cardioid. What is the ratio of the larger area to the smaller area? In general, what conclusion can you make about the relationship between the area bounded by a curve $r = 1 + \cos(θ)$ and the area bounded by a curve $r = a(1 + \cos(θ))$ , where $a$ is a constant?

11-93.

The spiral $r = e^{θ /10}$ is shown at right.

Calculate the slope of the curve at three points where the spiral crosses the ray $θ =\frac { \pi } { 3 }$ . What do you notice about the slope at these points?
Examine the graph at right. Notice that multiple coordinate points are plotted on the ray $θ =\frac { \pi } { 3 }$ . What is true about the slope at each of these points? Write a convincing argument to support your conclusion.

Your teacher will provide you with a model.

11-94.

No calculator! A particle moves in the plane so that at any time $t$ , $0 ≤ t ≤ 1$ , its position is given by $x(t) =\frac { 1 } { 4 }e^{8t} − 2t$ and $y(t) = e^{4t}$ . What are its velocity and speed when $t =\frac { 1 } { 4 }$ ? Homework Help ✎

Compute without a calculator

11-95.

Calculate the arc length of the parametric curve given by $x(t) = t^2 - t$ and $y(t) =\frac { 1 } { 3 }t^3 − t$ over $0 ≤ t ≤ 1$ . Homework Help ✎

11-96.

Multiple Choice: Which of the series below converge? Homework Help ✎

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n + 2 }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } n ! e ^ { - n }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } 3 \frac { n } { ( - 1 ) ^ { n } }$

I, III

I, II, III

11-97.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral ad briefly describe your method. Homework Help ✎

$\int _ { - 1 } ^ { 1 } x ^ { - 1 / 3 } d x$

$\int \cos(x) · e^{\sin(x)}dx$

$\int _ { - 6 } ^ { 6 } \operatorname { sin } ( x ^ { 3 } ) d x$

$\int \frac { \operatorname { cos } ( x ) } { \operatorname { sin } ( x ) } d x$

11-98.

The table below shows the total number of hamburgers sold at various times during the day for a popular fast food restaurant. Assume the function used to model the data in the table is a continuously increasing function. 11-98 HW eTool (Desmos). Homework Help ✎

Time	9 a.m.	11 a.m.	12 p.m.	1 p.m.	2 p.m.	4 p.m.	5 p.m.	7 p.m.
Hamburgers	$3$	$39$	$81$	$251$	$341$	$498$	$611$	$894$

At approximately what rate were hamburgers being sold at 10 a.m.? 4p.m.?
For what time interval was the rate of sales greatest?

11-99.

The population $P$ of mountain lions grows at a rate of $\frac { d P } { d t }=\frac { 1 } { 160 }P(40 − P)$ lions per year. At time $t = 0$ , there are $12$ lions. What is the population after three years? Homework Help ✎

11-100.

Multiple Choice: The graph of $f(x) = x –\frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 } - \frac { x ^ { 7 } } { 5040 } + \frac { x ^ { 9 } } { 362880 } - ...$ best approximates which function near $x = 0$ ? Homework Help ✎

$f(x) = x^2$

$f(x) = e^x$

$f(x) = \sin(x)$

$f(x) = \cos(x)$

11-101.

Multiple Choice: The interval of convergence for the series $\displaystyle \sum _ { n = 1 } ^ { \infty } \frac { ( x - 1 ) ^ { n } } { n ! }$ is: Homework Help ✎

$(−∞, ∞)$

$(−1, 1)$

$[−1, 1]$

$(−1, 1]$

$[−1, 1)$

11-102.

Multiple Choice: Let $M$ be the average value of $f(x) = 3 · 2^{x-1}$ over the interval $0 ≤ x ≤ 5$ . The $x$ -value for which $f(x) = M$ is: 11-102 HW eTool (Desmos). Homework Help ✎

$2.449$

$3.161$

$5.483$

$13.417$

$67.085$

Calculus Third Edition