Waste less time on Facebook — follow Brilliant.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Calculus
Taylor Series
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
Let \(f(x)\) be a function such that \(f(0)=3, f'(0)=1,\) and \(f(3)=1.\) Using a quadratic Taylor polynomial of \(f(x),\) we can approximate the value of \(f''(0)\) as \[f''(0) \approx -\frac{A}{B},\] where \(A\) and \(B\) are coprime integers. Find the value of \(AB.\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
by Brilliant Staff
Using the quadratic Taylor polynomial of \( f(x) = \frac{1}{x} \) at \( x=5, \) find the approximate value of \( \frac{1}{7} \) multiplied by \(5^3.\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
by Brilliant Staff
For the linear Taylor polynomial \(g(x)=ax+b\) of \(f(x)=-\frac{1}{x-1}\) at \(x=2,\) find the error \(e\) defined by \[e=\int_{2}^{3}{(g(x)-f(x))^2 dx}.\]
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
by Brilliant Staff
Using the quadratic Taylor polynomial of \( f(x) = \ln{x} \) at \( x=4, \) we can approximate the value of \( \ln{9}\) in the form \[\ln{A}+B, \] where \(A\) and \(B\) are rational numbers. Find the value of \( 2A^2B.\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
by Brilliant Staff
Let \(f(x)\) be a function such that \(f(0)=1, f'(0)=2,\) and \(f(3)=1.\) Using a quadratic Taylor polynomial of \(f(x),\) we can approximate the value of \(f''(0)\) as \[f''(0) \approx -\frac{A}{B},\] where \(A\) and \(B\) are coprime integers. Find the value of \(AB.\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Your answer seems reasonable.
Find out if you're right!
Sign up to access problem solutions.
That seems reasonable.
Find out if you're right!
by Brilliant Staff