11.3.1What is the slope of a polar curve?

Slopes of Polar Curves

11-76.

SLOPE OF A POLAR CURVE, Part One

Sketch the circle $r = 2\sin(θ)$ for $0 ≤ t ≤ θ$ .

Use the sketch to determine the slope of the curve at $θ = 1$ , $\frac { \pi } { 4 }$ , $\frac { \pi } { 2 }$ , and $\frac { 3 \pi } { 4 }$ . Explain how you determined your answers.
Convert the polar form of the circle $r = 2\sin(θ)$ to rectangular form and then write the derivative.
Confirm algebraically that your derivative from part (b) gives the predicted slopes of the curve at $θ = 0$ , $\frac { \pi } { 4 }$ , $\frac { \pi } { 2 }$ , and $\frac { 3 \pi } { 4 }$ from part (a).
Briefly summarize this method for determining the slope of a polar curve at a point.

11-77.

SLOPE OF A POLAR CURVE, Part Two

The spiral $r = θ$ for $θ ≥ 0$ is shown at right.

Try to rewrite $r = θ$ in rectangular form.
Your teammate is having difficulty rewriting $r = θ$ in rectangular form and suggests skipping this step. They say, “If we want to know the slope of the graph of $r = θ$ , let’s just find its derivative: $\frac { d r } { d \theta }$ .”
Think about your teammate’s suggestion. Is $\frac { d r } { d \theta }$ the same thing as $\frac { d y } { d x }$ ? Do they both represent a way to determine the slope of the spiral? Why or why not?

11-78.

The equations of many polar curves, like spirals, cannot be easily converted to rectilinear form. But the slope, $\frac { d y } { d x }$ , can still be found if you convert it into parametric form by using the identity:

$\left\{ \begin{array} { l } { x = r \operatorname { cos } ( \theta ) } \\ { y = r \operatorname { sin } ( \theta ) } \end{array} \right.$

Use the fact that $r = θ$ for the spiral to eliminate $r$ from the parametric function above.
Differentiate the parametric function from part (a) to get $\frac { d y } { d x }$ in terms of $θ$ .
Calculate the slope of the spiral for $θ=0,\frac{\pi}{2}$ , and $π$ . Use the graph above to verify that your answers for the slopes at these locations are reasonable.
Write a summary of this method to calculate the slope of a polar curve at a point.

11-79.

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

$r = 6$

$r = 3\sin(θ)$

$2 \sin(θ) + 5 \cos(θ) =\frac { 10 } { r }$

$r = (2θ)$

11-80.

Write the equation of the line tangent to the polar curve $r = -1 + \cos(θ) \text{ at } θ =\frac { \pi } { 2 }$ . Homework Help ✎

11-81.

Calculate the area of the region bounded by the curves $r = \left(2θ\right)$ and $r = –θ$ . A graph of this region is shown to the right.. 11-81 HW eTool (Desmos). Homework Help ✎

11-82.

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

$\int _ { 0 } ^ { 2 } \frac { 1 } { ( x - 2 ) ^ { 2 } } d x$

$\int \sec^2(x) \ln(\tan(x))dx$

$\int \frac { 3 } { ( x - 1 ) ( x + 2 ) } d x$

$\int 6x \tan(x^2)dx$

11-83.

Thoroughly investigate the graph of $y = xe^{x}$ . Identify all of the important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify any point(s) of inflection and intercepts and provide graphs of $y = f ′\left(x\right)$ and $y = f ″\left(x\right)$ . Be sure to justify all statements both graphically and analytically. 11-83 HW eTool (Desmos). Homework Help ✎

11-84.

An object is moving in a straight line such that its distance traveled after $t$ minutes is $s = –\ln( \frac { 1 } { t + 1 } )$ meters. Homework Help ✎

What is the object’s average velocity over $0 ≤ t ≤ 3$ ?
What is the acceleration of the object at $t = 2$ minutes?

11-85.

Multiple Choice: The graph of $y = f(x)$ is shown at right. The function is twice differentiable. Examine the graph and decide which statement below is true. Homework Help ✎

$f(2) < f ′(2) < f ″(2)$

$f(2) < f ″(2) < f ′(2)$

$f ′(2) < f(2) < f ″(2)$

$f ″(2) < f(2) < f ′(2)$

$f ″(2) < f ′(2) < f(2)$

Downward parabola, vertex at the point (2, comma 2), passing through the points (3, comma 1) & (1, comma 1).

11-86.

Multiple Choice: A particle moves on a plane curve so that at any time $t > 0$ its coordinates are given by $x = \left(2t - 1\right)^{4}$ , $y = t^{2} + 1$ . The acceleration vector of the particle at $t = 1$ is: Homework Help ✎

$〈1, 2〉$

$〈4, 2〉$

$〈8, 2〉$

$〈24, 2〉$

$〈48, 2〉$

11-87.

Multiple Choice: The slope of the curve $\frac { x } { x - 1 }+ xy = y^2 - 3x$ at $(2, 4)$ is: Homework Help ✎

$−1$

11-88.

Multiple Choice: The radius of convergence of $\displaystyle \sum _ { n = 1 } ^ { \infty } n ! x ^ { n }$ is: Homework Help ✎

$−1$

$\frac { 1 } { 2 }$

$\sqrt { 2 }$

11-89.

Multiple Choice: The base of a solid is the region between the curve $y = x \sin\left(x\right)$ and the $x$ -axis on the interval $0 ≤ x ≤ π$ . The cross-sections perpendicular to the $x$ -axis are semicircles. The volume of the solid is: Homework Help ✎

$1.127$

$1.271$

$1.721$

$2.171$

$2.711$

Calculus Third Edition