교과서 질문
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.
f(w) = w³ -w / w
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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 3.8.46a
45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
x³+y³=2xy; (1, 1)
검증된 단계별 안내1
First, substitute the given point (1, 1) into the equation of the curve x³ + y³ = 2xy to verify if it satisfies the equation.
Substitute x = 1 and y = 1 into the equation: (1)³ + (1)³ = 2(1)(1).
Calculate the left side of the equation: 1 + 1 = 2.
Calculate the right side of the equation: 2 * 1 * 1 = 2.
Since both sides of the equation are equal, the point (1, 1) lies on the curve.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
1m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In the context of the equation x³ + y³ = 2xy, we differentiate both sides with respect to x, treating y as a function of x. This allows us to find the derivative dy/dx, which is essential for determining the slope of the tangent line at a specific point.
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative
Tangent Line
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point. To find the equation of the tangent line, we use the point-slope form, which requires the slope (found via differentiation) and the coordinates of the point on the curve.
추천 영상:
가이드 코스
05:13Slopes of Tangent Lines
Verifying Points on Curves
Verifying that a point lies on a curve involves substituting the coordinates of the point into the equation of the curve. If the left-hand side equals the right-hand side after substitution, the point is confirmed to be on the curve. In this case, substituting (1, 1) into the equation x³ + y³ = 2xy helps establish that the point is indeed on the curve before proceeding to find the tangent line.
추천 영상:
11:41Summary of Curve Sketching
관련 실천
교과서 질문
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
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교과서 질문
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √2x+1; a= 4
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교과서 질문
Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>
a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)
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교과서 질문
Shrinking isosceles triangle The hypotenuse of an isosceles right triangle decreases in length at a rate of 4 m/s.
a. At what rate is the area of the triangle changing when the legs are 5 m long?
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교과서 질문
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
sin y = 5x⁴−5; (1, π)
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