교과서 질문
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
b. Graph the tangent lines on the given graph.
x+y³−y=1; x=1
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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
3장, 문제 3.8.13b
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.x⁴+y⁴ = 2;(1,−1)
검증된 단계별 안내1
Start by differentiating both sides of the equation with respect to x. The equation is x⁴ + y⁴ = 2. Use implicit differentiation, which involves differentiating y with respect to x as well.
Differentiate x⁴ with respect to x, which gives 4x³. For y⁴, apply the chain rule: differentiate y⁴ with respect to y to get 4y³, then multiply by dy/dx (the derivative of y with respect to x).
Set up the equation from the differentiation: 4x³ + 4y³(dy/dx) = 0. This equation represents the derivative of the original equation.
Solve for dy/dx, which represents the slope of the curve. Rearrange the equation to isolate dy/dx: dy/dx = -4x³ / 4y³.
Substitute the given point (1, -1) into the equation for dy/dx. Replace x with 1 and y with -1 to find the slope at this specific point.
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주요 개념
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Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for curves defined by equations like x⁴ + y⁴ = 2, where y cannot be easily isolated.
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative
Slope of a Curve
The slope of a curve at a given point represents the rate of change of the y-coordinate with respect to the x-coordinate at that point. Mathematically, it is found by evaluating the derivative of the function at the specified point. In the context of implicit differentiation, once we find dy/dx, we can substitute the coordinates of the point (1, -1) to determine the slope of the curve at that location.
추천 영상:
11:41Summary of Curve Sketching
Chain Rule
The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential in implicit differentiation, as it allows us to differentiate terms involving y when applying the derivative to both sides of an equation.
추천 영상:
05:02Intro to the Chain Rule
관련 실천
교과서 질문
62–65. {Use of Tech} Graphing f and f'
b. Compute and graph f'.
f(x) = (x−1) sin^−1 x on [−1,1]
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교과서 질문
{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.
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교과서 질문
A differential equation is an equation involving an unknown function and its derivatives. Consider the differential equation y′′(t)+y(t) = 0.
b. Show that y = B cos t satisfies the equation for any constant B.
254
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교과서 질문
Vertical tangent lines
b. Does the curve have any horizontal tangent lines? Explain.
426
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교과서 질문
109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.
b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.
242
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