Question

2. Consider the cardioid $r = 1 + \sin(\theta)$, $0 \le \theta \le 2\pi$. a) Find an expression for $\frac{dy}{dx}$ of the graph. What is the slope of the tangent line when $\theta = \frac{\pi}{3}$? b) Find the horizontal and vertical tangents of the graph. Hint: First show that $\cos^2(\theta) - \sin^2(\theta) = 1 - 2\sin^2(\theta)$.

          2. Consider the cardioid $r = 1 + \sin(\theta)$, $0 \le \theta \le 2\pi$.
a) Find an expression for $\frac{dy}{dx}$ of the graph. What is the slope of the tangent line
when $\theta = \frac{\pi}{3}$?
b) Find the horizontal and vertical tangents of the graph. Hint: First show that
$\cos^2(\theta) - \sin^2(\theta) = 1 - 2\sin^2(\theta)$.

2. Consider the cardioid r = 1 + sin(θ), 0 ≤θ≤ 2π.
a) Find an expression for (dy)/(dx) of the graph. What is the slope of the tangent line
when θ = (π)/(3)?
b) Find the horizontal and vertical tangents of the graph. Hint: First show that
cos^2(θ) - sin^2(θ) = 1 - 2sin^2(θ).

Added by Ruth G.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

06/25/2023

Step 1

We can use the following formula: $\frac{dy}{dx} = \frac{r'sin(\theta) + rcos(\theta)}{r'cos(\theta) - rsin(\theta)}$ where $r' = \frac{dr}{d\theta}$. Show more…

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2. Consider the cardioid $r = 1 + sin(\theta)$, $0 \leq \theta \leq 2\pi$. a) Find an expression for $\frac{dy}{dx}$ of the graph. What is the slope of the tangent line when $\theta = \frac{\pi}{3}$? b) Find the horizontal and vertical tangents of the graph. Hint: First show that $cos^2(\theta) - sin^2(\theta) = 1 - 2sin^2(\theta)$.