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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 3 - Derivatives문제 3.7.43
3장, 문제 3.7.43

The eight curve Find the slopes of the curve y⁴ = y² – x² at the two points shown here.


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To find the slope of the curve at a given point, we need to find the derivative of the curve equation with respect to x. The given equation is y⁴ = y² - x².
Differentiate both sides of the equation with respect to x. Use implicit differentiation: d/dx(y⁴) = d/dx(y² - x²).
Apply the chain rule to differentiate y⁴ and y²: 4y³(dy/dx) = 2y(dy/dx) - 2x.
Rearrange the equation to solve for dy/dx: dy/dx = (2x) / (4y³ - 2y).
Substitute the coordinates of the given points into the derivative to find the slope at each point. For the point (√3/4, √3/2), substitute x = √3/4 and y = √3/2. For the point (√3/4, 1/2), substitute x = √3/4 and y = 1/2.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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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 isolated. In this case, the equation y⁴ = y² - x² involves both x and y, making it necessary to apply implicit differentiation to find the slope of the curve at specific points. This method allows us to differentiate both sides of the equation with respect to x, treating y as a function of x.
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative

Slope of a Curve

The slope of a curve at a given point is defined as the derivative of the function at that point. It represents the rate of change of y with respect to x. For the curve defined by the equation y⁴ = y² - x², finding the slope at the specified points involves evaluating the derivative obtained from implicit differentiation and substituting the coordinates of the points into this derivative.
추천 영상:
05:45
Understanding Slope Fields

Critical Points

Critical points are values of x where the derivative of a function is either zero or undefined. In the context of the given curve, identifying critical points can help determine where the slope changes, which is essential for understanding the behavior of the curve. The points provided in the question are critical for evaluating the slope, as they represent specific locations on the curve where the slope needs to be calculated.
추천 영상:
04:50
Critical Points
관련 실천
교과서 질문

For what value or values of the constant m, if any, is


ƒ(x) = { sin 2x, x ≤ 0

{ mx, x > 0


a. continuous at x = 0?

b. differentiable at x = 0?


Give reasons for your answers.

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교과서 질문

Derivatives in Differential Form


In Exercises 17–28, find dy.


y = x√(1 − x²)

165
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교과서 질문

Differential Estimates of Change


In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.


The change in the lateral surface area S = 2πrh of a right circular cylinder when the height changes from h₀ to h₀ + dh and the radius does not change

138
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교과서 질문

Finding Derivative Values


In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.


f(u) = ((u − 1) / (u + 1))², u = g(x) = (1 / x²) − 1, x = −1

291
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교과서 질문

Second Derivatives


In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.


3 + sin y = y – x³

182
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교과서 질문

Free-Fall Applications

Free fall on Mars and Jupiter The equations for free fall at the surfaces of Mars and Jupiter (s in meters, t in seconds) are s = 1.86t² on Mars and s = 11.44t² on Jupiter. How long does it take a rock falling from rest to reach a velocity of 27.8 m/sec (about 100 km/h) on each planet?

287
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