Calculus Examples

f (t) = 4 \sqrt{t} - t

$f (t) = 4 \sqrt{t} - t$ ,

$[0, 16]$

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