Domanda del libro di testo
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
g(s) = 4s³ - 8s² +4s / 4s
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.7.106a
Deriving trigonometric identities
a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.
Guida verificata passo dopo passo1
Step 1: Differentiate the left side of the identity with respect to t. The left side is \( \cos(2t) \). Using the chain rule, the derivative is \(-2\sin(2t)\).
Step 2: Differentiate the right side of the identity with respect to t. The right side is \( \cos^2(t) - \sin^2(t) \). Use the chain rule and the power rule to differentiate each term separately.
Step 3: For \( \cos^2(t) \), use the chain rule: the derivative is \( 2\cos(t)(-\sin(t)) = -2\cos(t)\sin(t) \).
Step 4: For \( \sin^2(t) \), use the chain rule: the derivative is \( 2\sin(t)\cos(t) \).
Step 5: Combine the derivatives from steps 3 and 4: \(-2\cos(t)\sin(t) - 2\sin(t)\cos(t) = -2\sin(t)\cos(t) - 2\sin(t)\cos(t) = -4\sin(t)\cos(t)\). Set this equal to the derivative from step 1, \(-2\sin(2t)\), and simplify to show \( \sin(2t) = 2\sin(t)\cos(t) \).
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They serve as fundamental tools in calculus and can simplify complex expressions. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle identities, which are essential for manipulating and proving relationships between sine and cosine functions.
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Verifying Trig Equations as Identities
Differentiation
Differentiation is a fundamental concept in calculus that involves finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In the context of trigonometric functions, differentiation applies specific rules, such as the derivatives of sine and cosine, to derive new relationships. This process is crucial for proving identities by showing that two expressions have the same derivative.
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Percorso guidato
05:53Finding Differentials
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 sin(2t) = 2sin(t)cos(t) is derived from the sine and cosine functions. Understanding these formulas is essential for simplifying expressions and proving identities, as they provide a direct relationship between angles and their trigonometric values.
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Pratica correlata
Domanda del libro di testo
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(s) = 4s³+3s; a= -3, -1
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Domanda del libro di testo
79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>
a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)
x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)
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Domanda del libro di testo
Comparing velocities Two stones are thrown vertically upward, each with an initial velocity of 48 ft/s at time t=0. One stone is thrown from the edge of a bridge that is 32 feet above the ground, and the other stone is thrown from ground level. The height above the ground of the stone thrown from the bridge after t seconds is f(t) = − 16t²+48t+32. and the height of the stone thrown from the ground after t seconds is g(t) = −16t²+48t.
a. Show that the stones reach their high points at the same time.
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Domanda del libro di testo
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 1/x; P (1,1)
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Domanda del libro di testo
Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.
a. f(x) = (x-2)^1/3
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