Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Introduction to Trigonometric Functions

Calculus

0. Functions

Introduction to Trigonometric Functions

Previous problemNext problem

1:52 minutes

Problem 3.7.60a

Textbook Question

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 guidance

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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).

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.

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.

Watch next

Master Converting between Degrees & Radians with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Refer to the given figure. Write the radius r of the circle in terms of α and θ.

Textbook Question

In Exercises 39–42, express the given quantity in terms of sin x and cos x.

sin (2π − x)

Textbook Question

A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of x = 10 cos t, where x is measured in centimeters and t is measured in seconds. See the accompanying figure.

Find the spring’s displacement when t = 0, t = π/3, and t = 3π/4.

Textbook Question

Assume that a particle’s position on the x-axis is given by

x = 3 cos t + 4 sin t,

where x is measured in feet and t is measured in seconds.

a. Find the particle’s position when t = 0, t = π/2, and t = π.

Multiple Choice

Convert the angle $540°$ from degrees to radians.