Fundamental Theorem of Calculus (Part II)

Front

What is the Fundamental Theorem of Calculus (Part II)?

Back

It states that if F is an antiderivative of f on an interval [a, b], then \( \int_a^b f(x)dx = F(b) - F(a) \).

Front

How do you find the derivative of an integral function?

Back

Use the Fundamental Theorem of Calculus: if \( F(x) = \int_{a}^{g(x)} f(t) dt \), then \( F'(x) = f(g(x)) \cdot g'(x) \).

Front

What is the derivative of \( F(x) = \int_{e^2}^{\sqrt{x}} t dt \)?

Back

\( F'(x) = \sqrt{x} \cdot \frac{1}{2} x^{-\frac{1}{2}} \).

Front

What is the derivative of \( F(x) = \int_{\pi}^{g(x)} \frac{\sqrt{t^4 - t}}{\cos(t)} dt \)?

Back

\( F'(x) = \frac{\sqrt{g(x)^4 - g(x)}}{\cos(g(x))} g'(x) \).

Front

If \( f(x) = \frac{d}{dx} \int_7^{2x} t^3 dt \), what is \( f(-1) \)?

Back

\( f(-1) = -16 \).

Front

What is the derivative of \( F(x) = \int_3^x 6t^2 dt \)?

Back

\( F'(x) = 6x^2 \).

Front

How do you differentiate an integral with variable limits?

Back

Use the formula: \( F'(x) = f(h(x)) \cdot h'(x) - f(g(x)) \cdot g'(x) \) for \( F(x) = \int_{g(x)}^{h(x)} f(t) dt \).

Front

What is the significance of the Fundamental Theorem of Calculus?

Back

It connects differentiation and integration, showing that they are inverse processes.

Front

What is an antiderivative?

Back

An antiderivative of a function f is a function F such that \( F' = f \).

Front

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

Back

The definite integral \( \int_a^b f(x)dx \) represents the net area between the curve f(x) and the x-axis from x=a to x=b.