Textbook Question
49. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample:
c. ∫ v du = u·v - ∫ u dv
1
views
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- 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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.2.60c
60. Two Methods
c. Verify that your answers to parts (a) and (b) are consistent.
Verified step by step guidance1
Review the answers you obtained in parts (a) and (b) carefully, noting the expressions or values you found for the quantity or function in question.
Identify the key results from both parts, such as derivatives, integrals, or function values, depending on what parts (a) and (b) asked you to find.
Set the expressions from parts (a) and (b) equal to each other or compare them directly to check for consistency. This might involve simplifying both expressions to a common form.
If the expressions look different, try algebraic manipulation such as factoring, expanding, or using trigonometric identities to see if they can be shown to be equivalent.
Conclude that the answers are consistent if you can demonstrate that both methods lead to the same result or expression, confirming the correctness of your solutions.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Verification of Solutions
Verification involves checking that the solutions obtained from different methods satisfy the original problem or equation. This ensures consistency and correctness by substituting the solutions back into the initial conditions or equations.
Recommended video:
04:00
Solutions to Basic Differential Equations
Multiple Solution Methods
Using more than one method to solve a problem helps confirm the accuracy of results. Common methods in calculus include analytical techniques, graphical analysis, or numerical approximation, each providing a different perspective on the solution.
Recommended video:
07:33Euler's Method
Consistency in Mathematical Results
Consistency means that different approaches yield the same or compatible results, reinforcing the validity of the solution. It is a fundamental principle in mathematics to cross-check answers to avoid errors and ensure reliability.
Recommended video:
03:48Integrals Resulting in Natural Logs
Related Practice
Textbook Question
45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
46. ∫(0 to 2) x⁴ dx; n = 30
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.
Textbook Question
109. Escape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by W = ∫ from R to ∞ of F(x) dx, where R = 6370 km is the approximate radius of Earth, F(x) = (GMm)/x² is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of Earth, m is the mass of the object, and GM = 4 × 10¹⁴ m³/s².
c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light, c = 300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius R ≤ 2GM/c². For Earth to be a black hole, what would its radius need to be?
1
views
Textbook Question
Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.
c.
π
∫ sin(mx) cos(nx) dx = 0, when |m + n| is even
0
Textbook Question
3. What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following?
c. A factor of (x² + 2x + 6) in the denominator
1
views
Textbook Question
94. [Use of Tech] Skydiving A skydiver has a downward velocity given by v(t) = V_T [(1 - e^(-2gt/V_T))/(1 + e^(-2gt/V_T))],
where t = 0 is the instant the skydiver starts falling, g = 9.8 m/s² is the acceleration due to gravity, and V_T is the terminal velocity of the skydiver.
c. Verify by integration that the position function is given by
s(t) = V_T t + (V_T²/g) ln[(1 + e^(-2gt/V_T))/2],
where s'(t) = v(t) and s(0) = 0.
1
views