Calculus Examples

$f (t) = 4 \sqrt{t} - t$ , $[0, 16]$

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 $4 \sqrt{t} - t$ with respect to $t$ is $\frac{d}{d t} [4 \sqrt{t}] + \frac{d}{d t} [- t]$ .

$\frac{d}{d t} [4 \sqrt{t}] + \frac{d}{d t} [- t]$

Step 1.1.1.2

Evaluate $\frac{d}{d t} [4 \sqrt{t}]$ .

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Step 1.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\frac{d}{d t} [4 t^{\frac{1}{2}}] + \frac{d}{d t} [- t]$

Step 1.1.1.2.2

Since $4$ is constant with respect to $t$ , the derivative of $4 t^{\frac{1}{2}}$ with respect to $t$ is $4 \frac{d}{d t} [t^{\frac{1}{2}}]$ .

$4 \frac{d}{d t} [t^{\frac{1}{2}}] + \frac{d}{d t} [- t]$

Step 1.1.1.2.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 1.1.1.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.1.1.2.5

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 1.1.1.2.6

Combine the numerators over the common denominator.

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

Step 1.1.1.2.7

Simplify the numerator.

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Step 1.1.1.2.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.1.2.7.2

Subtract $2$ from $1$ .

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

Step 1.1.1.2.8

Move the negative in front of the fraction.

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

Step 1.1.1.2.9

Combine $\frac{1}{2}$ and $t^{- \frac{1}{2}}$ .

$4 \frac{t^{- \frac{1}{2}}}{2} + \frac{d}{d t} [- t]$

Step 1.1.1.2.10

Combine $4$ and $\frac{t^{- \frac{1}{2}}}{2}$ .

$\frac{4 t^{- \frac{1}{2}}}{2} + \frac{d}{d t} [- t]$

Step 1.1.1.2.11

Move $t^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{2 t^{\frac{1}{2}}} + \frac{d}{d t} [- t]$

Step 1.1.1.2.12

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2 t^{\frac{1}{2}}} + \frac{d}{d t} [- t]$

Step 1.1.1.2.13

Cancel the common factors.

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Step 1.1.1.2.13.1

Factor $2$ out of $2 t^{\frac{1}{2}}$ .

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

Step 1.1.1.2.13.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 t^{\frac{1}{2}}} + \frac{d}{d t} [- t]$

Step 1.1.1.2.13.3

Rewrite the expression.

$\frac{2}{t^{\frac{1}{2}}} + \frac{d}{d t} [- t]$

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$\frac{2}{t^{\frac{1}{2}}} - \frac{d}{d t} [t]$

Step 1.1.1.3.2

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

$\frac{2}{t^{\frac{1}{2}}} - 1 \cdot 1$

Step 1.1.1.3.3

Multiply $- 1$ by $1$ .

$f' (t) = \frac{2}{t^{\frac{1}{2}}} - 1$

Step 1.1.2

The first derivative of $f (t)$ with respect to $t$ is $\frac{2}{t^{\frac{1}{2}}} - 1$ .

$\frac{2}{t^{\frac{1}{2}}} - 1$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{2}{t^{\frac{1}{2}}} - 1 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{2}{t^{\frac{1}{2}}} - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$\frac{2}{t^{\frac{1}{2}}} = 1$

Step 1.2.3

Find the LCD of the terms in the equation.

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Step 1.2.3.1

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

$t^{\frac{1}{2}}, 1$

Step 1.2.3.2

The LCM of one and any expression is the expression.

$t^{\frac{1}{2}}$

Step 1.2.4

Multiply each term in $\frac{2}{t^{\frac{1}{2}}} = 1$ by $t^{\frac{1}{2}}$ to eliminate the fractions.

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Step 1.2.4.1

Multiply each term in $\frac{2}{t^{\frac{1}{2}}} = 1$ by $t^{\frac{1}{2}}$ .

$\frac{2}{t^{\frac{1}{2}}} t^{\frac{1}{2}} = 1 t^{\frac{1}{2}}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $t^{\frac{1}{2}}$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{2}{t^{\frac{1}{2}}} t^{\frac{1}{2}} = 1 t^{\frac{1}{2}}$

Step 1.2.4.2.1.2

Rewrite the expression.

$2 = 1 t^{\frac{1}{2}}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Multiply $t^{\frac{1}{2}}$ by $1$ .

$2 = t^{\frac{1}{2}}$

Step 1.2.5

Solve the equation.

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Step 1.2.5.1

Rewrite the equation as $t^{\frac{1}{2}} = 2$ .

$t^{\frac{1}{2}} = 2$

Step 1.2.5.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(t^{\frac{1}{2}})}^{2} = 2^{2}$

Step 1.2.5.3

Simplify the exponent.

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Step 1.2.5.3.1

Simplify the left side.

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Step 1.2.5.3.1.1

Simplify ${(t^{\frac{1}{2}})}^{2}$ .

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Step 1.2.5.3.1.1.1

Multiply the exponents in ${(t^{\frac{1}{2}})}^{2}$ .

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Step 1.2.5.3.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$t^{\frac{1}{2} \cdot 2} = 2^{2}$

Step 1.2.5.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.5.3.1.1.1.2.1

Cancel the common factor.

$t^{\frac{1}{2} \cdot 2} = 2^{2}$

Step 1.2.5.3.1.1.1.2.2

Rewrite the expression.

$t^{1} = 2^{2}$

Step 1.2.5.3.1.1.2

Simplify.

$t = 2^{2}$

Step 1.2.5.3.2

Simplify the right side.

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Step 1.2.5.3.2.1

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

$t = 4$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Convert expressions with fractional exponents to radicals.

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Step 1.3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{2}{\sqrt{t^{1}}} - 1$

Step 1.3.1.2

Anything raised to $1$ is the base itself.

$\frac{2}{\sqrt{t}} - 1$

Step 1.3.2

Set the denominator in $\frac{2}{\sqrt{t}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{t} = 0$

Step 1.3.3

Solve for $t$ .

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Step 1.3.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{t}}^{2} = 0^{2}$

Step 1.3.3.2

Simplify each side of the equation.

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Step 1.3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

${(t^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.3.3.2.2

Simplify the left side.

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Step 1.3.3.2.2.1

Simplify ${(t^{\frac{1}{2}})}^{2}$ .

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Step 1.3.3.2.2.1.1

Multiply the exponents in ${(t^{\frac{1}{2}})}^{2}$ .

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Step 1.3.3.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$t^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.3.3.2.2.1.1.2.1

Cancel the common factor.

$t^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.3.3.2.2.1.1.2.2

Rewrite the expression.

$t^{1} = 0^{2}$

Step 1.3.3.2.2.1.2

Simplify.

$t = 0^{2}$

Step 1.3.3.2.3

Simplify the right side.

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Step 1.3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$t = 0$

Step 1.3.4

Set the radicand in $\sqrt{t}$ less than $0$ to find where the expression is undefined.

$t < 0$

Step 1.3.5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$t \leq 0$

$(- \infty, 0]$

$t \leq 0$

$(- \infty, 0]$

Step 1.4

Evaluate $4 \sqrt{t} - t$ at each $t$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $t = 4$ .

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Step 1.4.1.1

Substitute $4$ for $t$ .

$4 \sqrt{4} - (4)$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Remove parentheses.

$4 \sqrt{4} - (4)$

Step 1.4.1.2.2

Simplify each term.

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Step 1.4.1.2.2.1

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

$4 \sqrt{2^{2}} - (4)$

Step 1.4.1.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$4 \cdot 2 - (4)$

Step 1.4.1.2.2.3

Multiply $4$ by $2$ .

$8 - (4)$

Step 1.4.1.2.2.4

Multiply $- 1$ by $4$ .

$8 - 4$

Step 1.4.1.2.3

Subtract $4$ from $8$ .

$4$

Step 1.4.2

Evaluate at $t = 0$ .

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Step 1.4.2.1

Substitute $0$ for $t$ .

$4 \sqrt{0} - (0)$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Remove parentheses.

$4 \sqrt{0} - (0)$

Step 1.4.2.2.2

Rewrite $0$ as $0^{2}$ .

$4 \sqrt{0^{2}} - (0)$

Step 1.4.2.2.3

Pull terms out from under the radical, assuming positive real numbers.

$4 \cdot 0 - (0)$

Step 1.4.2.2.4

Multiply $4$ by $0$ .

$0 - (0)$

Step 1.4.2.2.5

Subtract $0$ from $0$ .

$0$

Step 1.4.3

List all of the points.

$(4, 4), (0, 0)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $t = 0$ .

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Step 2.1.1

Substitute $0$ for $t$ .

$4 \sqrt{0} - (0)$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Remove parentheses.

$4 \sqrt{0} - (0)$

Step 2.1.2.2

Rewrite $0$ as $0^{2}$ .

$4 \sqrt{0^{2}} - (0)$

Step 2.1.2.3

Pull terms out from under the radical, assuming positive real numbers.

$4 \cdot 0 - (0)$

Step 2.1.2.4

Multiply $4$ by $0$ .

$0 - (0)$

Step 2.1.2.5

Subtract $0$ from $0$ .

$0$

Step 2.2

Evaluate at $t = 16$ .

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Step 2.2.1

Substitute $16$ for $t$ .

$4 \sqrt{16} - (16)$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Remove parentheses.

$4 \sqrt{16} - (16)$

Step 2.2.2.2

Simplify each term.

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Step 2.2.2.2.1

Rewrite $16$ as $4^{2}$ .

$4 \sqrt{4^{2}} - (16)$

Step 2.2.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$4 \cdot 4 - (16)$

Step 2.2.2.2.3

Multiply $4$ by $4$ .

$16 - (16)$

Step 2.2.2.2.4

Multiply $- 1$ by $16$ .

$16 - 16$

Step 2.2.2.3

Subtract $16$ from $16$ .

$0$

Step 2.3

List all of the points.

$(0, 0), (16, 0)$

Step 3

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

Absolute Maximum: $(4, 4)$

Absolute Minimum: $(0, 0), (16, 0)$

Step 4