Mathematics, 18.03.2021 01:50 thibeauxkristy

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Answer from: 1162

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

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Equality Properties

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

Algebra I

FunctionsFunction 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

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

Step 3: Solve

[Derivative] Substitute in function value: $\displaystyle 0 = 1 + 2cost$ [Subtraction Property of Equality] Isolate t term: $\displaystyle -1 = 2cost$ Rewrite: $\displaystyle 2cost = -1$ [Division Property of Equality] Isolate trig t term: $\displaystyle cost = \frac{-1}{2}$ [Equality Property] Inverse Trig: $\displaystyle t = cos^{-1}(\frac{-1}{2})$ 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

Answer from: Quest

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Answer from: Quest

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step-by-step explanation:

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