Calculus Examples

$f (x) = 5 \sqrt{x}$ , $[4, 9]$

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 $f (x) = 5 \sqrt{x}$ is continuous.

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Step 2.1

To find whether the function is continuous on $[4, 9]$ or not, find the domain of $f (x) = 5 \sqrt{x}$ .

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Step 2.1.1

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 2.1.2

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 2.2

$f (x)$ is continuous on $[4, 9]$ .

The function is continuous.

Step 3

Find the derivative.

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Step 3.1

Find the first derivative.

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Step 3.1.1

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

$\frac{d}{d x} [5 x^{\frac{1}{2}}]$

Step 3.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{1}{2}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$5 \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 3.1.3

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

$5 (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 3.1.4

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

$5 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 3.1.5

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

$5 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.1.6

Combine the numerators over the common denominator.

$5 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 3.1.7

Simplify the numerator.

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Step 3.1.7.1

Multiply $- 1$ by $2$ .

$5 (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 3.1.7.2

Subtract $2$ from $1$ .

$5 (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 3.1.8

Move the negative in front of the fraction.

$5 (\frac{1}{2} x^{- \frac{1}{2}})$

Step 3.1.9

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

$5 \frac{x^{- \frac{1}{2}}}{2}$

Step 3.1.10

Combine $5$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{5 x^{- \frac{1}{2}}}{2}$

Step 3.1.11

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

$f' (x) = \frac{5}{2 x^{\frac{1}{2}}}$

Step 3.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{5}{2 x^{\frac{1}{2}}}$ .

$\frac{5}{2 x^{\frac{1}{2}}}$

Step 4

Find if the derivative is continuous on $(4, 9)$ .

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Step 4.1

To find whether the function is continuous on $(4, 9)$ or not, find the domain of $f' (x) = \frac{5}{2 x^{\frac{1}{2}}}$ .

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Step 4.1.1

Convert expressions with fractional exponents to radicals.

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Step 4.1.1.1

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

$\frac{5}{2 \sqrt{x^{1}}}$

Step 4.1.1.2

Anything raised to $1$ is the base itself.

$\frac{5}{2 \sqrt{x}}$

Step 4.1.2

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 4.1.3

Set the denominator in $\frac{5}{2 \sqrt{x}}$ equal to $0$ to find where the expression is undefined.

$2 \sqrt{x} = 0$

Step 4.1.4

Solve for $x$ .

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Step 4.1.4.1

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

${(2 \sqrt{x})}^{2} = 0^{2}$

Step 4.1.4.2

Simplify each side of the equation.

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Step 4.1.4.2.1

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

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

Step 4.1.4.2.2

Simplify the left side.

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Step 4.1.4.2.2.1

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

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Step 4.1.4.2.2.1.1

Apply the product rule to $2 x^{\frac{1}{2}}$ .

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

Step 4.1.4.2.2.1.2

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

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

Step 4.1.4.2.2.1.3

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

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Step 4.1.4.2.2.1.3.1

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

$4 x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.1.4.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 4.1.4.2.2.1.3.2.1

Cancel the common factor.

$4 x^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.1.4.2.2.1.3.2.2

Rewrite the expression.

$4 x^{1} = 0^{2}$

Step 4.1.4.2.2.1.4

Simplify.

$4 x = 0^{2}$

Step 4.1.4.2.3

Simplify the right side.

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Step 4.1.4.2.3.1

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

$4 x = 0$

Step 4.1.4.3

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 4.1.4.3.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 4.1.4.3.2

Simplify the left side.

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Step 4.1.4.3.2.1

Cancel the common factor of $4$ .

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Step 4.1.4.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 4.1.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 4.1.4.3.3

Simplify the right side.

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Step 4.1.4.3.3.1

Divide $0$ by $4$ .

$x = 0$

Step 4.1.5

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 4.2

$f' (x)$ is continuous on $(4, 9)$ .

The function is continuous.

Step 5

The function is differentiable on $(4, 9)$ because the derivative is continuous on $(4, 9)$ .

The function is differentiable.

Step 6

$f (x)$ satisfies the two conditions for the mean value theorem. It is continuous on $[4, 9]$ and differentiable on $(4, 9)$ .

$f (x)$ is continuous on $[4, 9]$ and differentiable on $(4, 9)$ .

Step 7

Evaluate $f (a)$ from the interval $[4, 9]$ .

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Step 7.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = 5 \sqrt{4}$

Step 7.2

Simplify the result.

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Step 7.2.1

Remove parentheses.

$f (4) = 5 \sqrt{4}$

Step 7.2.2

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

$f (4) = 5 \sqrt{2^{2}}$

Step 7.2.3

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

$f (4) = 5 \cdot 2$

Step 7.2.4

Multiply $5$ by $2$ .

$f (4) = 10$

Step 7.2.5

The final answer is $10$ .

$10$

Step 8

Evaluate $f (b)$ from the interval $[4, 9]$ .

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Step 8.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = 5 \sqrt{9}$

Step 8.2

Simplify the result.

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Step 8.2.1

Remove parentheses.

$f (9) = 5 \sqrt{9}$

Step 8.2.2

Rewrite $9$ as $3^{2}$ .

$f (9) = 5 \sqrt{3^{2}}$

Step 8.2.3

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

$f (9) = 5 \cdot 3$

Step 8.2.4

Multiply $5$ by $3$ .

$f (9) = 15$

Step 8.2.5

The final answer is $15$ .

$15$

Step 9

Solve $\frac{5}{2 x^{\frac{1}{2}}} = \frac{- (f (b) + f (a))}{- (b + a)}$ for $x$ . $\frac{5}{2 x^{\frac{1}{2}}} = \frac{- (f (9) + f (4))}{- (9 + 4)}$ .

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Step 9.1

Factor each term.

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Step 9.1.1

Multiply $- 1$ by $10$ .

$\frac{5}{2 x^{\frac{1}{2}}} = \frac{15 - 10}{(9) - (4)}$

Step 9.1.2

Subtract $10$ from $15$ .

$\frac{5}{2 x^{\frac{1}{2}}} = \frac{5}{(9) - (4)}$

Step 9.1.3

Multiply $- 1$ by $4$ .

$\frac{5}{2 x^{\frac{1}{2}}} = \frac{5}{9 - 4}$

Step 9.1.4

Subtract $4$ from $9$ .

$\frac{5}{2 x^{\frac{1}{2}}} = \frac{5}{5}$

Step 9.1.5

Divide $5$ by $5$ .

$\frac{5}{2 x^{\frac{1}{2}}} = 1$

Step 9.2

Find the LCD of the terms in the equation.

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Step 9.2.1

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

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

Step 9.2.2

The LCM of one and any expression is the expression.

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

Step 9.3

Multiply each term in $\frac{5}{2 x^{\frac{1}{2}}} = 1$ by $2 x^{\frac{1}{2}}$ to eliminate the fractions.

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Step 9.3.1

Multiply each term in $\frac{5}{2 x^{\frac{1}{2}}} = 1$ by $2 x^{\frac{1}{2}}$ .

$\frac{5}{2 x^{\frac{1}{2}}} (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 9.3.2

Simplify the left side.

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Step 9.3.2.1

Rewrite using the commutative property of multiplication.

$2 \frac{5}{2 x^{\frac{1}{2}}} x^{\frac{1}{2}} = 1 (2 x^{\frac{1}{2}})$

Step 9.3.2.2

Cancel the common factor of $2$ .

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Step 9.3.2.2.1

Cancel the common factor.

$2 \frac{5}{2 x^{\frac{1}{2}}} x^{\frac{1}{2}} = 1 (2 x^{\frac{1}{2}})$

Step 9.3.2.2.2

Rewrite the expression.

$\frac{5}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 1 (2 x^{\frac{1}{2}})$

Step 9.3.2.3

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

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Step 9.3.2.3.1

Cancel the common factor.

$\frac{5}{x^{\frac{1}{2}}} x^{\frac{1}{2}} = 1 (2 x^{\frac{1}{2}})$

Step 9.3.2.3.2

Rewrite the expression.

$5 = 1 (2 x^{\frac{1}{2}})$

Step 9.3.3

Simplify the right side.

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Step 9.3.3.1

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

$5 = 2 x^{\frac{1}{2}}$

Step 9.4

Solve the equation.

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Step 9.4.1

Rewrite the equation as $2 x^{\frac{1}{2}} = 5$ .

$2 x^{\frac{1}{2}} = 5$

Step 9.4.2

Divide each term in $2 x^{\frac{1}{2}} = 5$ by $2$ and simplify.

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Step 9.4.2.1

Divide each term in $2 x^{\frac{1}{2}} = 5$ by $2$ .

$\frac{2 x^{\frac{1}{2}}}{2} = \frac{5}{2}$

Step 9.4.2.2

Simplify the left side.

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Step 9.4.2.2.1

Cancel the common factor.

$\frac{2 x^{\frac{1}{2}}}{2} = \frac{5}{2}$

Step 9.4.2.2.2

Divide $x^{\frac{1}{2}}$ by $1$ .

$x^{\frac{1}{2}} = \frac{5}{2}$

Step 9.4.3

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

${(x^{\frac{1}{2}})}^{2} = {(\frac{5}{2})}^{2}$

Step 9.4.4

Simplify the exponent.

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Step 9.4.4.1

Simplify the left side.

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Step 9.4.4.1.1

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

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Step 9.4.4.1.1.1

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

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Step 9.4.4.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = {(\frac{5}{2})}^{2}$

Step 9.4.4.1.1.1.2

Cancel the common factor of $2$ .

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Step 9.4.4.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\frac{5}{2})}^{2}$

Step 9.4.4.1.1.1.2.2

Rewrite the expression.

$x^{1} = {(\frac{5}{2})}^{2}$

Step 9.4.4.1.1.2

Simplify.

$x = {(\frac{5}{2})}^{2}$

Step 9.4.4.2

Simplify the right side.

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Step 9.4.4.2.1

Simplify ${(\frac{5}{2})}^{2}$ .

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Step 9.4.4.2.1.1

Apply the product rule to $\frac{5}{2}$ .

$x = \frac{5^{2}}{2^{2}}$

Step 9.4.4.2.1.2

Raise $5$ to the power of $2$ .

$x = \frac{25}{2^{2}}$

Step 9.4.4.2.1.3

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

$x = \frac{25}{4}$

Step 10

There is a tangent line found at $x = \frac{25}{4}$ parallel to the line that passes through the end points $a = 4$ and $b = 9$ .

There is a tangent line at $x = \frac{25}{4}$ parallel to the line that passes through the end points $a = 4$ and $b = 9$

Step 11