# Calculus Examples

,

If is continuous on the interval and differentiable on , then at least one real number exists in the interval such that . The mean value theorem expresses the relationship between the slope of the tangent to the curve at and the slope of the line through the points and .

If is continuous on

and if differentiable on ,

then there exists at least one point, in : .

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

is continuous on .

The function is continuous.

The function is continuous.

Find the first derivative.

Since is constant with respect to , the derivative of with respect to is .

By the Sum Rule, the derivative of with respect to is .

Differentiate using the Power Rule which states that is where .

Since is constant with respect to , the derivative of with respect to is .

Simplify the expression.

Add and to get .

Multiply by to get .

The derivative of with respect to is .

Since there are no variables in the expression, the domain is all real numbers.

is continuous on .

The function is continuous.

The function is continuous.

The function is differentiable on because the derivative is continuous on .

The function is differentiable.

satisfies the two conditions for the mean value theorem. It is continuous on and differentiable on .

is continuous on and differentiable on .

Replace the variable with in the expression.

Simplify the result.

Subtract from to get .

Multiply by to get .

The final answer is .

Replace the variable with in the expression.

Simplify the result.

Subtract from to get .

Multiply by to get .

The final answer is .

Rewrite the equation as .

Reduce the expression by cancelling the common factors.

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Move .

Cancel the common factor.

Rewrite the expression.

Simplify the numerator.

Multiply by to get .

Add and to get .

Simplify the denominator.

Factor out of .

Factor out of .

Factor out of .

Rewrite.

Add and to get .

Reduce the expression by cancelling the common factors.

Reduce the expression by cancelling the common factors.

Factor out of .

Factor out of .

Factor out of .

Move .

Multiply by to get .

Cancel the common factor.

Rewrite the expression.

Simplify the expression.

Multiply by to get .

Divide by to get .

Since , the equation will always be true.

Always true

Always true

The graph is a straight line. There is a tangent line at every on the curve, which is parallel to the line that passes through the end points and .

There is a tangent line at every x on the curve, which is parallel to the line that passes through the end points and