Textbook Question
Find an equation of the line tangent to the following curves at the given value of x.
y = 4 sin x cos x; x = π/3
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 3.8.63b
Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.
Verified step by step guidance1
First, rewrite the given equation y(x² + 4) = 8 in terms of y: y = 8 / (x² + 4). This represents the Witch of Agnesi curve.
To find the tangent lines, we need to determine the derivative of y with respect to x. Use the quotient rule for differentiation: if y = u/v, then y' = (u'v - uv') / v².
Apply the quotient rule to y = 8 / (x² + 4). Here, u = 8 and v = x² + 4. Calculate u' = 0 and v' = 2x. Substitute these into the quotient rule formula to find y'.
Set y = 1 in the original equation to find the x-values where the curve has y = 1. Solve 1(x² + 4) = 8 for x, which simplifies to x² + 4 = 8, leading to x² = 4. Solve for x to find x = ±2.
Substitute x = ±2 into the derivative y' to find the slope of the tangent lines at these points. Use the point-slope form of a line, y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point of tangency, to write the equations of the tangent lines.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Implicit Differentiation
Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y. In this case, since y is expressed in terms of x², we differentiate both sides of the equation with respect to x, applying the chain rule where necessary. This allows us to find dy/dx, which is essential for determining the slope of the tangent lines.
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Finding The Implicit Derivative
Tangent Line Equation
The equation of a tangent line at a given point on a curve can be expressed using the point-slope form: y - y₀ = m(x - x₀), where (x₀, y₀) is the point of tangency and m is the slope at that point. To find the tangent lines when y=1, we first need to determine the corresponding x-values and then calculate the slope using the derivative obtained from implicit differentiation.
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Witch of Agnesi
The Witch of Agnesi is a specific type of curve defined by the equation y(x² + 4) = 8, which can be rearranged to express y in terms of x. This curve is notable in calculus for its symmetrical properties and its applications in probability and statistics. Understanding its shape and behavior is crucial for analyzing tangent lines and their equations.
Related Practice
Textbook Question
An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.
Find the velocity of the oscillator, v(t) =y′(t).
Textbook Question
{Use of Tech} Hours of daylight The number of hours of daylight at any point on Earth fluctuates throughout the year. In the Northern Hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At 40° north latitude, the length of a day is approximated by D(t) = 12−3 cos (2π(t+10) / 365), where D is measured in hours and 0≤t≤365 is measured in days, with t=0 corresponding to January 1.
b. Find the rate at which the daylight function changes.
Textbook Question
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 1/x+1; a = -1/2;5
Textbook Question
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
Textbook Question
62–65. {Use of Tech} Graphing f and f'
b. Compute and graph f'.
f(x)=e^−x tan^−1 x on [0,∞)