Calculus Examples

$f (x) = \frac{x^{2} - 3 x - 4}{x + 2}$ , $[- 1, 4]$

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 $f (x) = \frac{x^{2} - 3 x - 4}{x + 2}$ is continuous.

Tap for more steps...

Step 2.1

To find whether the function is continuous on $[- 1, 4]$ or not, find the domain of $f (x) = \frac{x^{2} - 3 x - 4}{x + 2}$ .

Tap for more steps...

Step 2.1.1

Set the denominator in $\frac{x^{2} - 3 x - 4}{x + 2}$ equal to $0$ to find where the expression is undefined.

$x + 2 = 0$

Step 2.1.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.1.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 2) \cup (- 2, \infty)$

Set-Builder Notation:

${x | x \neq - 2}$

Interval Notation:

$(- \infty, - 2) \cup (- 2, \infty)$

Set-Builder Notation:

${x | x \neq - 2}$

Step 2.2

$f (x)$ is continuous on $[- 1, 4]$ .

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

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} - 3 x - 4$ and $g (x) = x + 2$ .

$\frac{(x + 2) \frac{d}{d x} [x^{2} - 3 x - 4] - (x^{2} - 3 x - 4) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 3.1.2

Differentiate.

Tap for more steps...

Step 3.1.2.1

By the Sum Rule, the derivative of $x^{2} - 3 x - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 4]$ .

$\frac{(x + 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 4]) - (x^{2} - 3 x - 4) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

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 $n = 2$ .

$\frac{(x + 2) (2 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 4]) - (x^{2} - 3 x - 4) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 3.1.2.3

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

$\frac{(x + 2) (2 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]) - (x^{2} - 3 x - 4) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 3.1.2.4

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

$\frac{(x + 2) (2 x - 3 \cdot 1 + \frac{d}{d x} [- 4]) - (x^{2} - 3 x - 4) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 3.1.2.5

Multiply $- 3$ by $1$ .

$\frac{(x + 2) (2 x - 3 + \frac{d}{d x} [- 4]) - (x^{2} - 3 x - 4) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 3.1.2.6

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{(x + 2) (2 x - 3 + 0) - (x^{2} - 3 x - 4) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 3.1.2.7

Add $2 x - 3$ and $0$ .

$\frac{(x + 2) (2 x - 3) - (x^{2} - 3 x - 4) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 3.1.2.8

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

$\frac{(x + 2) (2 x - 3) - (x^{2} - 3 x - 4) (\frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 3.1.2.9

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

$\frac{(x + 2) (2 x - 3) - (x^{2} - 3 x - 4) (1 + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 3.1.2.10

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{(x + 2) (2 x - 3) - (x^{2} - 3 x - 4) (1 + 0)}{{(x + 2)}^{2}}$

Step 3.1.2.11

Simplify the expression.

Tap for more steps...

Step 3.1.2.11.1

Add $1$ and $0$ .

$\frac{(x + 2) (2 x - 3) - (x^{2} - 3 x - 4) \cdot 1}{{(x + 2)}^{2}}$

Step 3.1.2.11.2

Multiply $- 1$ by $1$ .

$\frac{(x + 2) (2 x - 3) - (x^{2} - 3 x - 4)}{{(x + 2)}^{2}}$

Step 3.1.3

Simplify.

Tap for more steps...

Step 3.1.3.1

Apply the distributive property.

$\frac{(x + 2) (2 x - 3) - x^{2} - (- 3 x) - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2

Simplify the numerator.

Tap for more steps...

Step 3.1.3.2.1

Simplify each term.

Tap for more steps...

Step 3.1.3.2.1.1

Expand $(x + 2) (2 x - 3)$ using the FOIL Method.

Tap for more steps...

Step 3.1.3.2.1.1.1

Apply the distributive property.

$\frac{x (2 x - 3) + 2 (2 x - 3) - x^{2} - (- 3 x) - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.1.1.2

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 3 + 2 (2 x - 3) - x^{2} - (- 3 x) - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.1.1.3

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 3 + 2 (2 x) + 2 \cdot - 3 - x^{2} - (- 3 x) - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 3.1.3.2.1.2.1

Simplify each term.

Tap for more steps...

Step 3.1.3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x \cdot x + x \cdot - 3 + 2 (2 x) + 2 \cdot - 3 - x^{2} - (- 3 x) - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.1.2.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.1.3.2.1.2.1.2.1

Move $x$ .

$\frac{2 (x \cdot x) + x \cdot - 3 + 2 (2 x) + 2 \cdot - 3 - x^{2} - (- 3 x) - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot - 3 + 2 (2 x) + 2 \cdot - 3 - x^{2} - (- 3 x) - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.1.2.1.3

Move $- 3$ to the left of $x$ .

$\frac{2 x^{2} - 3 \cdot x + 2 (2 x) + 2 \cdot - 3 - x^{2} - (- 3 x) - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.1.2.1.4

Multiply $2$ by $2$ .

$\frac{2 x^{2} - 3 x + 4 x + 2 \cdot - 3 - x^{2} - (- 3 x) - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.1.2.1.5

Multiply $2$ by $- 3$ .

$\frac{2 x^{2} - 3 x + 4 x - 6 - x^{2} - (- 3 x) - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.1.2.2

Add $- 3 x$ and $4 x$ .

$\frac{2 x^{2} + x - 6 - x^{2} - (- 3 x) - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.1.3

Multiply $- 3$ by $- 1$ .

$\frac{2 x^{2} + x - 6 - x^{2} + 3 x - - 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.1.4

Multiply $- 1$ by $- 4$ .

$\frac{2 x^{2} + x - 6 - x^{2} + 3 x + 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.2

Subtract $x^{2}$ from $2 x^{2}$ .

$\frac{x^{2} + x - 6 + 3 x + 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.3

Add $x$ and $3 x$ .

$\frac{x^{2} + 4 x - 6 + 4}{{(x + 2)}^{2}}$

Step 3.1.3.2.4

Add $- 6$ and $4$ .

$f' (x) = \frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}}$

Step 3.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}}$ .

$\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}}$

Step 4

Find if the derivative is continuous on $(- 1, 4)$ .

Tap for more steps...

Step 4.1

To find whether the function is continuous on $(- 1, 4)$ or not, find the domain of $f' (x) = \frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}}$ .

Tap for more steps...

Step 4.1.1

Set the denominator in $\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x + 2)}^{2} = 0$

Step 4.1.2

Solve for $x$ .

Tap for more steps...

Step 4.1.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 4.1.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.1.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 2) \cup (- 2, \infty)$

Set-Builder Notation:

${x | x \neq - 2}$

Interval Notation:

$(- \infty, - 2) \cup (- 2, \infty)$

Set-Builder Notation:

${x | x \neq - 2}$

Step 4.2

$f' (x)$ is continuous on $(- 1, 4)$ .

The function is continuous.

Step 5

The function is differentiable on $(- 1, 4)$ because the derivative is continuous on $(- 1, 4)$ .

The function is differentiable.

Step 6

$f (x)$ satisfies the two conditions for the mean value theorem. It is continuous on $[- 1, 4]$ and differentiable on $(- 1, 4)$ .

$f (x)$ is continuous on $[- 1, 4]$ and differentiable on $(- 1, 4)$ .

Step 7

Evaluate $f (a)$ from the interval $[- 1, 4]$ .

Tap for more steps...

Step 7.1

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

$f (- 1) = \frac{{(- 1)}^{2} - 3 \cdot - 1 - 4}{(- 1) + 2}$

Step 7.2

Simplify the result.

Tap for more steps...

Step 7.2.1

Simplify the numerator.

Tap for more steps...

Step 7.2.1.1

Raise $- 1$ to the power of $2$ .

$f (- 1) = \frac{1 - 3 \cdot - 1 - 4}{- 1 + 2}$

Step 7.2.1.2

Multiply $- 3$ by $- 1$ .

$f (- 1) = \frac{1 + 3 - 4}{- 1 + 2}$

Step 7.2.1.3

Add $1$ and $3$ .

$f (- 1) = \frac{4 - 4}{- 1 + 2}$

Step 7.2.1.4

Subtract $4$ from $4$ .

$f (- 1) = \frac{0}{- 1 + 2}$

Step 7.2.2

Simplify the expression.

Tap for more steps...

Step 7.2.2.1

Add $- 1$ and $2$ .

$f (- 1) = \frac{0}{1}$

Step 7.2.2.2

Divide $0$ by $1$ .

$f (- 1) = 0$

Step 7.2.3

The final answer is $0$ .

$0$

Step 8

Solve $\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}} = \frac{- (f (b) + f (a))}{- (b + a)}$ for $x$ . $\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}} = \frac{- (f (4) + f (- 1))}{- (4 - 1)}$ .

Tap for more steps...

Step 8.1

Factor each term.

Tap for more steps...

Step 8.1.1

Multiply $- 1$ by $0$ .

$\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}} = \frac{0 + 0}{(4) - (- 1)}$

Step 8.1.2

Add $0$ and $0$ .

$\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}} = \frac{0}{(4) - (- 1)}$

Step 8.1.3

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}} = \frac{0}{4 + 1}$

Step 8.1.4

Add $4$ and $1$ .

$\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}} = \frac{0}{5}$

Step 8.1.5

Divide $0$ by $5$ .

$\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}} = 0$

Step 8.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 8.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

${(x + 2)}^{2}, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

${(x + 2)}^{2}$

Step 8.3

Multiply each term in $\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}} = 0$ by ${(x + 2)}^{2}$ to eliminate the fractions.

Tap for more steps...

Step 8.3.1

Multiply each term in $\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}} = 0$ by ${(x + 2)}^{2}$ .

$\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}} {(x + 2)}^{2} = 0 {(x + 2)}^{2}$

Step 8.3.2

Simplify the left side.

Tap for more steps...

Step 8.3.2.1

Cancel the common factor of ${(x + 2)}^{2}$ .

Tap for more steps...

Step 8.3.2.1.1

Cancel the common factor.

$\frac{x^{2} + 4 x - 2}{{(x + 2)}^{2}} {(x + 2)}^{2} = 0 {(x + 2)}^{2}$

Step 8.3.2.1.2

Rewrite the expression.

$x^{2} + 4 x - 2 = 0 {(x + 2)}^{2}$

Step 8.3.3

Simplify the right side.

Tap for more steps...

Step 8.3.3.1

Multiply $0$ by ${(x + 2)}^{2}$ .

$x^{2} + 4 x - 2 = 0$

Step 8.4

Solve the equation.

Tap for more steps...

Step 8.4.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 8.4.2

Substitute the values $a = 1$ , $b = 4$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 2)}}{2 \cdot 1}$

Step 8.4.3

Simplify.

Tap for more steps...

Step 8.4.3.1

Simplify the numerator.

Tap for more steps...

Step 8.4.3.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 2}}{2 \cdot 1}$

Step 8.4.3.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

Tap for more steps...

Step 8.4.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 2}}{2 \cdot 1}$

Step 8.4.3.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{- 4 \pm \sqrt{16 + 8}}{2 \cdot 1}$

Step 8.4.3.1.3

Add $16$ and $8$ .

$x = \frac{- 4 \pm \sqrt{24}}{2 \cdot 1}$

Step 8.4.3.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

Tap for more steps...

Step 8.4.3.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 4 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 8.4.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 8.4.3.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 8.4.3.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{6}}{2}$

Step 8.4.3.3

Simplify $\frac{- 4 \pm 2 \sqrt{6}}{2}$ .

$x = - 2 \pm \sqrt{6}$

Step 8.4.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 8.4.4.1

Simplify the numerator.

Tap for more steps...

Step 8.4.4.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 2}}{2 \cdot 1}$

Step 8.4.4.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

Tap for more steps...

Step 8.4.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 2}}{2 \cdot 1}$

Step 8.4.4.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{- 4 \pm \sqrt{16 + 8}}{2 \cdot 1}$

Step 8.4.4.1.3

Add $16$ and $8$ .

$x = \frac{- 4 \pm \sqrt{24}}{2 \cdot 1}$

Step 8.4.4.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

Tap for more steps...

Step 8.4.4.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 4 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 8.4.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 8.4.4.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 8.4.4.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{6}}{2}$

Step 8.4.4.3

Simplify $\frac{- 4 \pm 2 \sqrt{6}}{2}$ .

$x = - 2 \pm \sqrt{6}$

Step 8.4.4.4

Change the $\pm$ to $+$ .

$x = - 2 + \sqrt{6}$

Step 8.4.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 8.4.5.1

Simplify the numerator.

Tap for more steps...

Step 8.4.5.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 2}}{2 \cdot 1}$

Step 8.4.5.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

Tap for more steps...

Step 8.4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 2}}{2 \cdot 1}$

Step 8.4.5.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{- 4 \pm \sqrt{16 + 8}}{2 \cdot 1}$

Step 8.4.5.1.3

Add $16$ and $8$ .

$x = \frac{- 4 \pm \sqrt{24}}{2 \cdot 1}$

Step 8.4.5.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

Tap for more steps...

Step 8.4.5.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 4 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 8.4.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 8.4.5.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 8.4.5.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{6}}{2}$

Step 8.4.5.3

Simplify $\frac{- 4 \pm 2 \sqrt{6}}{2}$ .

$x = - 2 \pm \sqrt{6}$

Step 8.4.5.4

Change the $\pm$ to $-$ .

$x = - 2 - \sqrt{6}$

Step 8.4.6

The final answer is the combination of both solutions.

$x = - 2 + \sqrt{6}, - 2 - \sqrt{6}$

Step 9

There is a tangent line found at $x = - 2 + \sqrt{6}$ parallel to the line that passes through the end points $a = - 1$ and $b = 4$ .

There is a tangent line at $x = - 2 + \sqrt{6}$ parallel to the line that passes through the end points $a = - 1$ and $b = 4$

Step 10

There is a tangent line found at $x = - 2 - \sqrt{6}$ parallel to the line that passes through the end points $a = - 1$ and $b = 4$ .

There is a tangent line at $x = - 2 - \sqrt{6}$ parallel to the line that passes through the end points $a = - 1$ and $b = 4$

Step 11