Question

please someone helpme

Answer 1

Answer:

$\displaystyle t = \frac{2 \pi}{3}, \ \frac{4 \pi}{3}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right  

Equality Properties

• Multiplication Property of Equality
• Division Property of Equality
• Addition Property of Equality
• Subtraction Property of Equality

Algebra I

• Functions
• Function Notation
• [Interval Notation] - [Brackets] imply inclusive, (Parenthesis) imply exclusive

Pre-Calculus

• Unit Circle

Calculus

Derivatives

Derivative Notation

The definition of a derivative is the slope of the tangent line

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

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

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

Trig Derivative:                                                                                                        $\displaystyle \frac{d}{dx}[sinu] = u'cosu$

Step-by-step explanation:

Step 1: Define

[Given] q(t) = t + 2sint

[Given] Interval [0, 2π]

[Solve] q'(t) = 0

• Horizontal tangent line have a slope of 0

Step 2: Differentiate

1. [Derivative] Basic Power Rule/Trig Derivative [Derivative Prop - Add]:      $\displaystyle q'(t) = 1 \cdot t^{1 - 1} + 1 \cdot 2cost$
2. [Derivative] Simplify exponent:                                                                     $\displaystyle q'(t) = 1 \cdot t^{0} + 1 \cdot 2cost$
3. [Derivative] Evaluate exponent:                                                                    $\displaystyle q'(t) = 1 \cdot 1 + 1 \cdot 2cost$
4. [Derivative] Multiply:                                                                                       $\displaystyle q'(t) = 1 + 2cost$

Step 3: Solve

1. [Derivative] Substitute in function value:                                                      $\displaystyle 0 = 1 + 2cost$
2. [Subtraction Property of Equality] Isolate t term:                                          $\displaystyle -1 = 2cost$
3. Rewrite:                                                                                                           $\displaystyle 2cost = -1$
4. [Division Property of Equality] Isolate trig t term:                                         $\displaystyle cost = \frac{-1}{2}$
5. [Equality Property] Inverse Trig:                                                                    $\displaystyle t = cos^{-1}(\frac{-1}{2})$
6. Evaluate [Unit Circle, Interval]:                                                                       $\displaystyle t = \frac{2 \pi}{3}, \ \frac{4 \pi}{3}$

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

Unit: Derivatives

Book: College Calculus 10e