6.6-6.8b Warm up

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 \( \int_a^b f(x)dx = F(b) - F(a) \).

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 \( b \).

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.

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.

Front

Evaluate \( \int_0^x g(t) dt \) when \( g(t) = 2t \).

Back

\( \int_0^x 2t dt = [t^2]_0^x = x^2 \).

Front

What is the value of \( \int_{-1}^3 f'(x) dx \)?

Back

By the Fundamental Theorem of Calculus, \( \int_{-1}^3 f'(x) dx = f(3) - f(-1) \).

Front

If \( f(x) = \int_0^x g(t) dt \), what is \( f'(x) \)?

Back

By the Fundamental Theorem of Calculus, \( f'(x) = g(x) \).

Front

What does \( f''(x) \) represent?

Back

The second derivative \( f''(x) \) represents the rate of change of the rate of change of the function \( f \), indicating the concavity of the function.

Front

Evaluate \( \int_{-1}^3 f''(x) dx \).

Back

By the Fundamental Theorem of Calculus, \( \int_{-1}^3 f''(x) dx = f'(3) - f'(-1) \).

Front

What is the significance of the area under the curve in integration?

Back

The area under the curve of a function from \( a \) to \( b \) represents the total accumulation of the quantity represented by the function over that interval.