Question

Calculus question. Urgent help needed. Due today.

Answer 1

Answer:

12  (none of these)

Step-by-step explanation:

To find the slope at a point, we need to find the derivative of the function at the point, then evaluate at that point

derivative with respect to theta ( r = 6 cos 2theta - 5) evaluated at theta = pi/4

dr/dtheta  = 6  * dr dtheta (cos 2theta -5)

We need to find the derivative of cos 2 theta

d /dtheta = -2 sin 2 theta

dr/dtheta  = 6  *  (-2sin (2theta) -0)

dr/dtheta = -12 sin (2theta)  evaluated at theta= pi/4

dr/dtheta = -12 sin (2*pi/4)

dr/ dtheta = -12 sin (pi/2)

                = -12(-1)

                  =12

Answer 2

Answer:

None of these

General Formulas and Concepts:

Pre-Calculus

• Unit Circle

Calculus

Derivatives

Derivative Notation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:                                                                                 $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Trig Derivative:                                                                                                       $\displaystyle \frac{d}{dx}[cos(u)] = -u'sin(u)$

Polar Derivative:                                                                                                     $\displaystyle \frac{dy}{dx} = \frac{rcos(\theta) + r'sin(\theta)}{r'cos(\theta) - rsin(\theta)}$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle r = 6cos(2 \theta) - 5$

$\displaystyle \theta = \frac{\pi}{4}$

Step 2: Differentiate

1. Trig Derivative [Chain Rule]:                                                                         $\displaystyle r' = \frac{d}{d\theta}[6cos(2 \theta) - 5] \cdot \frac{d}{d\theta}[2\theta]$
2. Rewrite [Derivative Rule - Subtraction]:                                                       $\displaystyle r' = \bigg[ \frac{d}{d\theta}[6cos(2 \theta)] - \frac{d}{d\theta}[5] \bigg] \cdot \frac{d}{d\theta}[2\theta]$
3. Rewrite [Derivative Rule - Multiplied Constant]:                                         $\displaystyle r' = \bigg[ 6\frac{d}{d\theta}[cos(2 \theta)] - \frac{d}{d\theta}[5] \bigg] \cdot \frac{d}{d\theta}[2\theta]$
4. Trig Derivative:                                                                                             $\displaystyle r' = [-6sin(2\theta)] \cdot \frac{d}{d\theta}[2\theta]$
5. Basic Power Rule:                                                                                         $\displaystyle r' = -6sin(2\theta) \cdot 2\theta^{1 - 1}$
6. Simplify:                                                                                                         $\displaystyle r' = -12sin(2\theta)$
7. Substitute in variables [Polar Derivative]:                                                   $\displaystyle \frac{dy}{dx} = \frac{[6cos(2\theta) - 5]cos(\theta) + [-12sin(2\theta)]sin(\theta)}{[-12sin(2\theta)]cos(\theta) - [6cos(2\theta) - 5]sin(\theta)}$
8. [Polar Derivative] Simplify:                                                                           $\displaystyle \frac{dy}{dx} = \frac{-[6cos(\theta)cos(2\theta) - 12sin(\theta)sin(2\theta) - 5cos(\theta)]}{5sin(\theta)cos(2\theta) + 12cos(\theta)sin(2\theta) - 5sin(\theta)}$

Step 3: Find Slope

1. Substitute in θ [Polar Derivative]:                                                                 $\displaystyle \frac{dy}{dx} \bigg| \limits_\bigg{\theta = \frac{\pi}{4}} = \frac{-[6cos\frac{\pi}{4})cos(2 \cdot \frac{\pi}{4}) - 12sin(\frac{\pi}{4})sin(2 \cdot \frac{\pi}{4}) - 5cos(\frac{\pi}{4})]}{5sin(\frac{\pi}{4})cos(2 \cdot \frac{\pi}{4}) + 12cos(\frac{\pi}{4})sin(2 \cdot \frac{\pi}{4}) - 5sin(\frac{\pi}{4})}$
2. Evaluate [Unit Circle]:                                                                                   $\displaystyle \frac{dy}{dx} \bigg| \limits_\bigg{\theta = \frac{\pi}{4}} = \frac{17}{7}$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Polar Derivatives and Area (BC Only)

Book: College Calculus 10e