Textbook Question
The general polynomial of degree n has the form
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,
where aₙ ≠ 0. Find P'(x).
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 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)
Verified step by step guidance1
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.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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).
Recommended video:
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.
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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.
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11:41Summary of Curve Sketching
Related Practice
Textbook Question
If the original 24 m edge length x of a cube decreases at the rate of 5 m/min, when x = 3 m at what rate does the cube’s
a. surface area change?
Textbook Question
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
Textbook Question
Assume that f'(3) = −1, g'(2) = 5, g(2) = 3, and y = f(g(x)). What is y' at x = 2?
Textbook Question
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?
Textbook Question
Find y⁽⁴⁾ = d⁴y/dx⁴ if:
a. y = −2 sin x