Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. -Each antiderivative of an nth-degree polynomial function is an (n+1)th-degree polynomial function.

Solution

Verified

Step 1

1 of 2

Assume we have the polynomial of degree nn which can be written as

Pn=an xn+an1 xn1+an2 xn2++a1 x+a0P_n=a_n\ x^n+a_{n-1}\ x^{n-1}+a_{n-2}\ x^{n-2}+\dots+a_1\ x+ a_0

and now calculate its anti-derivative using sum and power rules

Pn dx=an xn+1n+1+an1 xnn+an2 xn1n1++a1 x22+a0x+C\int P_n\ dx=a_n\ \frac{x^{n+1}}{n+1}+a_{n-1}\ \frac{x^{n}}{n}+a_{n-2}\ \frac{x^{n-1}}{n-1}+\dots+a_1\ \frac{x^2}2+ a_0x+C

which can be written as follows

Pn+1 dx=bn+1 xn+1+bn xn+bn1 xn1++b2 x2+b1 x+b0P_{n+1}\ dx=b_{n+1}\ x^{n+1}+b_{n}\ x^{n}+b_{n-1}\ x^{n-1}+\dots+b_2\ x^2+ b_1\ x+b_0

Hence the statement is true

Create an account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy
Continue with GoogleContinue with Facebook

Create an account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy
Continue with GoogleContinue with Facebook

Recommended textbook solutions

Calculus 10th Edition by Bruce H. Edwards, Ron Larson

Calculus

10th EditionBruce H. Edwards, Ron Larson
11,084 solutions
Calculus: Early Transcendentals 8th Edition by James Stewart

Calculus: Early Transcendentals

8th EditionJames Stewart
11,070 solutions
Calculus: Early Transcendentals 9th Edition by Daniel K. Clegg, James Stewart, Saleem Watson

Calculus: Early Transcendentals

9th EditionDaniel K. Clegg, James Stewart, Saleem Watson
11,044 solutions
The Practice of Statistics for the AP Exam 5th Edition by Daniel S. Yates, Daren S. Starnes, David Moore, Josh Tabor

The Practice of Statistics for the AP Exam

5th EditionDaniel S. Yates, Daren S. Starnes, David Moore, Josh Tabor
2,433 solutions

Related questions