11/23- Warmup

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What is the Fundamental Theorem of Calculus Part 2?

Back

The Fundamental Theorem of Calculus Part 2 states that if F is an antiderivative of f on an interval [a, b], then \( \int_a^b f(x)dx = F(b) - F(a) \).

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How do you find the value of a function at a point using its derivative?

Back

To find the value of a function at a point using its derivative, you can use the Fundamental Theorem of Calculus, which relates the integral of the derivative to the change in the function's values.

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What is the area under a curve represented by an integral?

Back

The area under a curve represented by an integral \( \int_a^b f(x)dx \) is the net area between the curve and the x-axis from x=a to x=b.

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How do you evaluate the integral \( \int_0^2 f(x)dx \) if f(x) is given?

Back

To evaluate the integral \( \int_0^2 f(x)dx \), you need to find the antiderivative F(x) of f(x) and then compute \( F(2) - F(0) \).

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What is the significance of the semicircle in the context of derivatives?

Back

The semicircle in the context of derivatives can represent the graph of a function's derivative, indicating the slope of the function at various points.

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How do you express the area of a semicircle with radius r?

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The area of a semicircle with radius r is given by the formula \( \frac{1}{2} \pi r^2 \).

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What is the relationship between definite integrals and the area under a curve?

Back

Definite integrals calculate the net area under a curve, taking into account the regions above and below the x-axis.

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How do you find the value of f(3) given f'(x) and f(1)?

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To find f(3), use the Fundamental Theorem of Calculus to integrate f' from 1 to 3 and add the value f(1) to the result.

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What does it mean to leave an answer as a fraction?

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Leaving an answer as a fraction means expressing the result in the form of a ratio of two integers, rather than as a decimal or mixed number.

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How do you evaluate the integral \( \int_0^2 \frac{3x^2-1}{x^2}dx \)?

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To evaluate the integral, simplify the integrand and then find the antiderivative before applying the limits of integration.