Textbook Question
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(x) = 4x²+1; a= 2,4
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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.
Verified step by step guidance1
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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Guided course
5:04Converting between Degrees & Radians
Related Practice
Textbook Question
Vertical tangent lines
b. Does the curve have any horizontal tangent lines? Explain.
Textbook Question
{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.
b. Graph the volume function and verify that the tank is empty when t=1000 min.
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)
Textbook Question
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
sin y = 5x⁴−5; (1, π)
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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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