Calculus Examples

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

Step 1

Find the critical points.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1.1

Differentiate.

Tap for more steps...

Step 1.1.1.1.1

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

$\frac{d}{d x} [x] + \frac{d}{d x} [e^{- 4 x}]$

Step 1.1.1.1.2

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

$1 + \frac{d}{d x} [e^{- 4 x}]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [e^{- 4 x}]$ .

Tap for more steps...

Step 1.1.1.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - 4 x$ .

Tap for more steps...

Step 1.1.1.2.1.1

To apply the Chain Rule, set $u$ as $- 4 x$ .

$1 + \frac{d}{d u} [e^{u}] \frac{d}{d x} [- 4 x]$

Step 1.1.1.2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$1 + e^{u} \frac{d}{d x} [- 4 x]$

Step 1.1.1.2.1.3

Replace all occurrences of $u$ with $- 4 x$ .

$1 + e^{- 4 x} \frac{d}{d x} [- 4 x]$

Step 1.1.1.2.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

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

Step 1.1.1.2.3

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

$1 + e^{- 4 x} (- 4 \cdot 1)$

Step 1.1.1.2.4

Multiply $- 4$ by $1$ .

$1 + e^{- 4 x} \cdot - 4$

Step 1.1.1.2.5

Move $- 4$ to the left of $e^{- 4 x}$ .

$f' (x) = 1 - 4 e^{- 4 x}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $1 - 4 e^{- 4 x}$ .

$1 - 4 e^{- 4 x}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $1 - 4 e^{- 4 x} = 0$ .

Tap for more steps...

Step 1.2.1

Set the first derivative equal to $0$ .

$1 - 4 e^{- 4 x} = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

$- 4 e^{- 4 x} = - 1$

Step 1.2.3

Divide each term in $- 4 e^{- 4 x} = - 1$ by $- 4$ and simplify.

Tap for more steps...

Step 1.2.3.1

Divide each term in $- 4 e^{- 4 x} = - 1$ by $- 4$ .

$\frac{- 4 e^{- 4 x}}{- 4} = \frac{- 1}{- 4}$

Step 1.2.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{- 4 e^{- 4 x}}{- 4} = \frac{- 1}{- 4}$

Step 1.2.3.2.1.2

Divide $e^{- 4 x}$ by $1$ .

$e^{- 4 x} = \frac{- 1}{- 4}$

Step 1.2.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.1

Dividing two negative values results in a positive value.

$e^{- 4 x} = \frac{1}{4}$

Step 1.2.4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- 4 x}) = \ln (\frac{1}{4})$

Step 1.2.5

Expand the left side.

Tap for more steps...

Step 1.2.5.1

Expand $\ln (e^{- 4 x})$ by moving $- 4 x$ outside the logarithm.

$- 4 x \ln (e) = \ln (\frac{1}{4})$

Step 1.2.5.2

The natural logarithm of $e$ is $1$ .

$- 4 x \cdot 1 = \ln (\frac{1}{4})$

Step 1.2.5.3

Multiply $- 4$ by $1$ .

$- 4 x = \ln (\frac{1}{4})$

Step 1.2.6

Divide each term in $- 4 x = \ln (\frac{1}{4})$ by $- 4$ and simplify.

Tap for more steps...

Step 1.2.6.1

Divide each term in $- 4 x = \ln (\frac{1}{4})$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{\ln (\frac{1}{4})}{- 4}$

Step 1.2.6.2

Simplify the left side.

Tap for more steps...

Step 1.2.6.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{\ln (\frac{1}{4})}{- 4}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{4})}{- 4}$

Step 1.2.6.3

Simplify the right side.

Tap for more steps...

Step 1.2.6.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{1}{4})}{4}$

Step 1.3

Find the values where the derivative is undefined.

Tap for more steps...

Step 1.3.1

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.

Step 1.4

Evaluate $x + e^{- 4 x}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 1.4.1

Evaluate at $x = - \frac{\ln (\frac{1}{4})}{4}$ .

Tap for more steps...

Step 1.4.1.1

Substitute $- \frac{\ln (\frac{1}{4})}{4}$ for $x$ .

$(- \frac{\ln (\frac{1}{4})}{4}) + e^{- 4 (- \frac{\ln (\frac{1}{4})}{4})}$

Step 1.4.1.2

Simplify each term.

Tap for more steps...

Step 1.4.1.2.1

Rewrite $\frac{\ln (\frac{1}{4})}{4}$ as $\frac{1}{4} \ln (\frac{1}{4})$ .

$- (\frac{1}{4} \ln (\frac{1}{4})) + e^{- 4 (- \frac{\ln (\frac{1}{4})}{4})}$

Step 1.4.1.2.2

Simplify $\frac{1}{4} \ln (\frac{1}{4})$ by moving $\frac{1}{4}$ inside the logarithm.

$- \ln ({(\frac{1}{4})}^{\frac{1}{4}}) + e^{- 4 (- \frac{\ln (\frac{1}{4})}{4})}$

Step 1.4.1.2.3

Apply the product rule to $\frac{1}{4}$ .

$- \ln (\frac{1^{\frac{1}{4}}}{4^{\frac{1}{4}}}) + e^{- 4 (- \frac{\ln (\frac{1}{4})}{4})}$

Step 1.4.1.2.4

One to any power is one.

$- \ln (\frac{1}{4^{\frac{1}{4}}}) + e^{- 4 (- \frac{\ln (\frac{1}{4})}{4})}$

Step 1.4.1.2.5

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.4.1.2.5.1

Move the leading negative in $- \frac{\ln (\frac{1}{4})}{4}$ into the numerator.

$- \ln (\frac{1}{4^{\frac{1}{4}}}) + e^{- 4 \frac{- \ln (\frac{1}{4})}{4}}$

Step 1.4.1.2.5.2

Factor $4$ out of $- 4$ .

$- \ln (\frac{1}{4^{\frac{1}{4}}}) + e^{4 (- 1) \frac{- \ln (\frac{1}{4})}{4}}$

Step 1.4.1.2.5.3

Cancel the common factor.

$- \ln (\frac{1}{4^{\frac{1}{4}}}) + e^{4 \cdot - 1 \frac{- \ln (\frac{1}{4})}{4}}$

Step 1.4.1.2.5.4

Rewrite the expression.

$- \ln (\frac{1}{4^{\frac{1}{4}}}) + e^{- 1 (- \ln (\frac{1}{4}))}$

Step 1.4.1.2.6

Multiply $- 1$ by $- 1$ .

$- \ln (\frac{1}{4^{\frac{1}{4}}}) + e^{1 \ln (\frac{1}{4})}$

Step 1.4.1.2.7

Multiply $\ln (\frac{1}{4})$ by $1$ .

$- \ln (\frac{1}{4^{\frac{1}{4}}}) + e^{\ln (\frac{1}{4})}$

Step 1.4.1.2.8

Exponentiation and log are inverse functions.

$- \ln (\frac{1}{4^{\frac{1}{4}}}) + \frac{1}{4}$

Step 1.4.2

List all of the points.

$(- \frac{\ln (\frac{1}{4})}{4}, - \ln (\frac{1}{4^{\frac{1}{4}}}) + \frac{1}{4})$

Step 2

Evaluate at the included endpoints.

Tap for more steps...

Step 2.1

Evaluate at $x = - 2$ .

Tap for more steps...

Step 2.1.1

Substitute $- 2$ for $x$ .

$(- 2) + e^{- 4 \cdot - 2}$

Step 2.1.2

Multiply $- 4$ by $- 2$ .

$- 2 + e^{8}$

Step 2.2

Evaluate at $x = 3$ .

Tap for more steps...

Step 2.2.1

Substitute $3$ for $x$ .

$(3) + e^{- 4 \cdot 3}$

Step 2.2.2

Simplify each term.

Tap for more steps...

Step 2.2.2.1

Multiply $- 4$ by $3$ .

$3 + e^{- 12}$

Step 2.2.2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 + \frac{1}{e^{12}}$

Step 2.3

List all of the points.

$(- 2, - 2 + e^{8}), (3, 3 + \frac{1}{e^{12}})$

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(- 2, - 2 + e^{8})$

Absolute Minimum: $(- \frac{\ln (\frac{1}{4})}{4}, - \ln (\frac{1}{4^{\frac{1}{4}}}) + \frac{1}{4})$

Step 4