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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.7.106b
Chapter 3, Problem 3.7.106b

Deriving trigonometric identities
b. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.

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1
Step 1: Start with the given identity for \( \cos 2t \), which is \( \cos 2t = 2 \cos^2 t - 1 \).
Step 2: Differentiate both sides of the equation with respect to \( t \). For the left side, use the derivative of \( \cos 2t \), which is \(-2 \sin 2t\) using the chain rule.
Step 3: For the right side, differentiate \( 2 \cos^2 t - 1 \). Use the chain rule to differentiate \( 2 \cos^2 t \), which gives \( 2 \times 2 \cos t \times (-\sin t) = -4 \cos t \sin t \). The derivative of \(-1\) is 0.
Step 4: Set the derivatives from both sides equal to each other: \(-2 \sin 2t = -4 \cos t \sin t\).
Step 5: Simplify the equation \(-2 \sin 2t = -4 \cos t \sin t\) to verify the identity for \( \sin 2t \). Recognize that \( \sin 2t = 2 \sin t \cos t \), confirming the identity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle identities. Understanding these identities is crucial for simplifying expressions and solving equations in trigonometry.
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Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of a function with respect to its variable. In the context of trigonometric functions, differentiation allows us to find the slopes of curves and can be used to verify identities by showing that two expressions yield the same derivative.
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Double Angle Formulas

Double angle formulas are specific trigonometric identities that express trigonometric functions of double angles in terms of single angles. For example, the formula for cosine of double angle is cos(2t) = 2cos²(t) - 1. These formulas are essential for simplifying expressions and proving identities, particularly when differentiating or integrating trigonometric functions.
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Related Practice
Textbook Question

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>


{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.


b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach 90% of the carrying capacity?

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Textbook Question

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = − 0.01x²+40x+100, 0≤x≤1500, a=1000

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31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).

b. Determine the instantaneous velocity of the projectile at t=1 and t = 2 seconds.

s(t)= −16t²+100t

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Textbook Question

58–59. Carry out the following steps.

b. Find the slope of the curve at the given point.

xy^5/2+x^3/2y=12; (4, 1)

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21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(t) = 1/√t; a=9, 1/4

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Textbook Question

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

b. How fast will the city be growing when it reaches a size of 38 mi²?

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