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/cos²u, u = g(x) = πx, x = 1/4
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
3장, 문제 3.7.60a
Computer Explorations
Use a CAS to perform the following steps in Exercises 55–62.
a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.
xy³ + tan(x + y) = 1, P(π/4, 0)
검증된 단계별 안내1
Start by understanding the given equation: \(xy^3 + \tan(x + y) = 1\). This is an implicit equation involving both x and y.
Use a Computer Algebra System (CAS) to plot the equation. Input the equation into the implicit plotter feature of the CAS to visualize the curve represented by the equation.
To verify if the point P(\(\frac{\pi}{4}, 0\)) satisfies the equation, substitute \(x = \frac{\pi}{4}\) and \(y = 0\) into the equation.
Calculate the left-hand side of the equation with the substituted values: \(\left(\frac{\pi}{4}\right)(0)^3 + \tan\left(\frac{\pi}{4} + 0\right)\).
Check if the calculated value from the previous step equals 1, which is the right-hand side of the equation. If it does, then the point P satisfies the equation.
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 Plotting
Implicit plotting involves graphing equations where the relationship between variables is not explicitly solved for one variable in terms of the others. In this case, the equation xy³ + tan(x + y) = 1 is plotted using a Computer Algebra System (CAS) to visualize solutions that satisfy the equation, including checking specific points like P(π/4, 0).
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative
Computer Algebra System (CAS)
A Computer Algebra System (CAS) is software designed to perform symbolic mathematical computations. It can handle tasks such as algebraic simplifications, solving equations, and plotting graphs. In this exercise, a CAS is used to plot the implicit equation and verify if the point P satisfies the equation, showcasing its utility in handling complex mathematical operations.
추천 영상:
05:25Determine Continuity Algebraically
Verification of Points on Curves
Verifying if a point lies on a curve involves substituting the point's coordinates into the equation and checking if the equation holds true. For the equation xy³ + tan(x + y) = 1, substituting P(π/4, 0) checks if the left-hand side equals 1, confirming whether P is a solution to the equation. This process is crucial for understanding the relationship between the equation and specific points.
추천 영상:
11:41Summary of Curve Sketching
관련 실천
교과서 질문
Assume that f'(3) = −1, g'(2) = 5, g(2) = 3, and y = f(g(x)). What is y' at x = 2?
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교과서 질문
In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.
y = (4 − 3x)⁹
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교과서 질문
The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s
a. surface area?
168
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교과서 질문
Find y⁽⁴⁾ = d⁴y/dx⁴ if:
a. y = −2 sin x
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교과서 질문
Differentiability and Continuity on an Interval
Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be
a. differentiable?
Give reasons for your answers.
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