Calculus Examples

$f (x) = e^{x} - x$ , $- 2 \leq x \leq 2$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

Multiply $- 1$ by $1$ .

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

Step 1.1.2

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

$e^{x} - 1$

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

$e^{x} - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$e^{x} = 1$

Step 1.2.3

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

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

Step 1.2.4

Expand the left side.

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Step 1.2.4.1

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

$x \ln (e) = \ln (1)$

Step 1.2.4.2

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

$x \cdot 1 = \ln (1)$

Step 1.2.4.3

Multiply $x$ by $1$ .

$x = \ln (1)$

Step 1.2.5

The natural logarithm of $1$ is $0$ .

$x = 0$

Step 1.3

Find the values where the derivative is undefined.

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

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Step 1.4.1

Evaluate at $x = 0$ .

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Step 1.4.1.1

Substitute $0$ for $x$ .

$e^{0} - (0)$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Anything raised to $0$ is $1$ .

$1 - (0)$

Step 1.4.1.2.2

Subtract $0$ from $1$ .

$1$

Step 1.4.2

List all of the points.

$(0, 1)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 2$ .

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Step 2.1.1

Substitute $- 2$ for $x$ .

$e^{- 2} - (- 2)$

Step 2.1.2

Simplify each term.

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Step 2.1.2.1

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

$\frac{1}{e^{2}} - (- 2)$

Step 2.1.2.2

Multiply $- 1$ by $- 2$ .

$\frac{1}{e^{2}} + 2$

Step 2.2

Evaluate at $x = 2$ .

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Step 2.2.1

Substitute $2$ for $x$ .

$e^{2} - (2)$

Step 2.2.2

Multiply $- 1$ by $2$ .

$e^{2} - 2$

Step 2.3

List all of the points.

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

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: $(2, e^{2} - 2)$

Absolute Minimum: $(0, 1)$

Step 4