6.6-6.8b Warm up

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Mathematics
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12th Grade
•
Hard
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1.
FLASHCARD QUESTION
Front
What is the Fundamental Theorem of Calculus?
Back
The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if \( f \) is continuous on \([a, b]\) and \( F \) is an antiderivative of \( f \), then \( \int_a^b f(x)dx = F(b) - F(a) \).
2.
FLASHCARD QUESTION
Front
Define the integral of a function.
Back
The integral of a function \( f(x) \) over an interval \([a, b]\) is the limit of the Riemann sums of \( f \) as the partition of the interval becomes infinitely fine, representing the area under the curve of \( f \) from \( a \) to \( b \).
3.
FLASHCARD QUESTION
Front
What does \( f'(x) \) represent in calculus?
Back
The derivative \( f'(x) \) represents the rate of change of the function \( f \) at the point \( x \), or the slope of the tangent line to the curve at that point.
4.
FLASHCARD QUESTION
Front
What is the relationship between a function and its integral?
Back
The integral of a function \( f(x) \) gives the accumulated area under the curve of \( f \) from a starting point to a given point, while the derivative \( f'(x) \) gives the instantaneous rate of change of that area.
5.
FLASHCARD QUESTION
Front
Evaluate \( \int_0^x g(t) dt \) when \( g(t) = 2t \).
Back
\( \int_0^x 2t dt = [t^2]_0^x = x^2 \).
6.
FLASHCARD QUESTION
Front
What is the value of \( \int_{-1}^3 f'(x) dx \)?
Back
By the Fundamental Theorem of Calculus, \( \int_{-1}^3 f'(x) dx = f(3) - f(-1) \).
7.
FLASHCARD QUESTION
Front
If \( f(x) = \int_0^x g(t) dt \), what is \( f'(x) \)?
Back
By the Fundamental Theorem of Calculus, \( f'(x) = g(x) \).
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