Interactive: Mean Value Theorem for Integrals

Mean Value Theorem for Integrals: If \(f\) is continuous on \([a,b]\), then there exists a number \(c\) in \([a,b]\) such that \[\int_a^b {f\left( x \right)dx = f\left( c \right)\left( {b - a} \right)} \]

Instructions: Drag a and b to see c calculated dynamically. The area under the curve from a to b is the same as the area of the pink rectangle formed by b - a and f(c).

Interactive: Net Area

Net Area: If the function f takes on both positive and negative values, then the Riemann sum is the result of the area under the curve above the x-axis less the area under the x-axis above the curve.

Instructions: Drag a and b to see net area change sign. Purple indicates positive net area and pink indicates negative net area.

Interactive: Definite Integral

Definite Integral: If \(f\) is a continuous function defined on the interval \([a, b]\) that is divided into \(n\) equal subintervals \(\Delta x = \frac{{b - a}}{n}\) where \(x_i^*\) lies in the ith subinterval then the definite integral of \(f\) from \(a\) to \(b\) follows:

\[\int_a^b {f\left( x \right)dx} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {f\left( {x_i^*} \right)} \,\Delta x\]

Instructions: Drag the slider to see that the area under the curve is the limit of Riemann sums. You can also change the sums to left, middle, right, and trapezoidal below the graph.

Riemann Sum: Left Middle Right Trapezoidal

Interactive: Derivative of Cosine

Derivative of the Sine Function: \[\frac{d}{{dx}}\left( {\cos x} \right) = - \sin x\]

Instructions: Drag P along \(f(x) = cos(x)\) to see that the slope of the tangent line through it traces out \(f'(x) = -sin(x)\).

Interactive: Fermat's Theorem

Fermat's Theorem: If \(f\) is a local maximum or minimum at \(c\), and if \(f'(c)\) exists, then \(f'(c)=0\).

Critical Number: A number \(c\) in the domain of \(f\) such that either \(f'(c)=0\) or \(f'(c)\) does not exist.

Instructions: With the mouse, move point P along the function and you will see it's derivative traced in green. Here c and d are the critical numbers for the function graphed in blue. Refresh browser to start over.

Interactive: Mean Value Theorem

Mean Value Theorem: Let \(f\) be a function that satisfies the following:

• \(f\) is continuous on a closed interval \([a, b]\).
• \(f\) is differentiable on the open interval \((a,b)\).

Then there is a number \(p\) in \((a,b)\) such that
\[f'\left( p \right) = \frac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\]

Instructions: With the mouse, move points A and B along the function to see the Mean Value Theorem in action. Refresh browser to start over.

YouTube Video Lectures by Rob Shone

Part 1: Continuous and Differentiable Functions (Part 1 of 3)

Part 2: Continuous and Differentiable Functions (Part 2 of 3)

Part 3: Continuous and Differentiable Functions (Part 3 of 3)

Interactive: Derivative of a Function

Derivative of a Function at \(a\), if it exists:

\[f'\left( a \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\]

Instructions: Drag the green dot representing a+h toward a to simulate h tending toward zero.

Interactive: Tangent Line at a Point

Definition: The tangent line to the curve \(y = f(x)\) at the point \(P(a, f(a))\) is the line through \(P\) with slope

\[m = \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) - f\left( a \right)}}{{x - a}}\]

provided that this limit exists.

Instructions: With the mouse, move the x-value toward a to see that the tangent line is the limiting position of the secant line shown dashed. You can also move the points on the function. Refresh browser to start over.