Calculus Examples

$y = 3 x^{3} - 2 x$ , $(1, 1)$

Step 1

If $f$ is continuous on the interval $[a, b]$ and differentiable on $(a, b)$ , then at least one real number $c$ exists in the interval $(a, b)$ such that $f' (c) = \frac{f (b) - f a}{b - a}$ . The mean value theorem expresses the relationship between the slope of the tangent to the curve at $x = c$ and the slope of the line through the points $(a, f (a))$ and $(b, f (b))$ .

If $f (x)$ is continuous on $[a, b]$

and if $f (x)$ differentiable on $(a, b)$ ,

then there exists at least one point, $c$ in $[a, b]$ : $f' (c) = \frac{f (b) - f a}{b - a}$ .

Step 2

Check if $f (x) = 3 x^{3} - 2 x$ is continuous.

Tap for more steps...

Step 2.1

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2.2

$f (x)$ is continuous on $[1, 1]$ .

The function is continuous.

Step 3

Find the derivative.

Tap for more steps...

Step 3.1

Find the first derivative.

Tap for more steps...

Step 3.1.1

By the Sum Rule, the derivative of $3 x^{3} - 2 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [- 2 x]$ .

$\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [- 2 x]$

Step 3.1.2

Evaluate $\frac{d}{d x} [3 x^{3}]$ .

Tap for more steps...

Step 3.1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x^{3}]$ .

$3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2 x]$

Step 3.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 (3 x^{2}) + \frac{d}{d x} [- 2 x]$

Step 3.1.2.3

Multiply $3$ by $3$ .

$9 x^{2} + \frac{d}{d x} [- 2 x]$

Step 3.1.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

Tap for more steps...

Step 3.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$9 x^{2} - 2 \frac{d}{d x} [x]$

Step 3.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$9 x^{2} - 2 \cdot 1$

Step 3.1.3.3

Multiply $- 2$ by $1$ .

$f' (x) = 9 x^{2} - 2$

Step 3.2

The first derivative of $f (x)$ with respect to $x$ is $9 x^{2} - 2$ .

$9 x^{2} - 2$

Step 4

Find if the derivative is continuous on No solution.

Tap for more steps...

Step 4.1

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4.2

$f' (x)$ is continuous on $No solution$ .

The function is continuous.

Step 5

The function is differentiable on $No solution$ because the derivative is continuous on $No solution$ .

The function is differentiable.

Step 6

$f (x)$ satisfies the two conditions for the mean value theorem. It is continuous on $[1, 1]$ and differentiable on $No solution$ .

No solution

Step 7

Evaluate $f (a)$ from the interval $[1, 1]$ .

Tap for more steps...

Step 7.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 3 {(1)}^{3} - 2 \cdot 1$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

One to any power is one.

$f (1) = 3 \cdot 1 - 2 \cdot 1$

Step 7.2.1.2

Multiply $3$ by $1$ .

$f (1) = 3 - 2 \cdot 1$

Step 7.2.1.3

Multiply $- 2$ by $1$ .

$f (1) = 3 - 2$

Step 7.2.2

Subtract $2$ from $3$ .

$f (1) = 1$

Step 7.2.3

The final answer is $1$ .

$1$

Step 8

The equation has an undefined fraction.

Undefined

Step 9

There are no solution, so there is no $x$ value where the tangent line is parallel to the line that passes through the end points $a = 1$ and $b = 1$ .

No x value found where the tangent line at x is parallel to the line that passes through the end points $a = 1$ and $b = 1$

Step 10