Calculus Examples

$g (t) = \frac{t}{\sqrt{5 + t^{2}}}$ , $[2, 5]$

Step 1

To find the average value of a function, the function should be continuous on the closed interval $[a, b]$ . To find whether $g (t)$ is continuous on $[2, 5]$ or not, find the domain of $g (t) = \frac{t}{\sqrt{5 + t^{2}}}$ .

Tap for more steps...

Step 1.1

Set the radicand in $\sqrt{5 + t^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$5 + t^{2} \geq 0$

Step 1.2

Solve for $t$ .

Tap for more steps...

Step 1.2.1

Subtract $5$ from both sides of the inequality.

$t^{2} \geq - 5$

Step 1.2.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 1.3

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

$\sqrt{5 + t^{2}} = 0$

Step 1.4

Solve for $t$ .

Tap for more steps...

Step 1.4.1

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

${\sqrt{5 + t^{2}}}^{2} = 0^{2}$

Step 1.4.2

Simplify each side of the equation.

Tap for more steps...

Step 1.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 + t^{2}}$ as ${(5 + t^{2})}^{\frac{1}{2}}$ .

${({(5 + t^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.4.2.2

Simplify the left side.

Tap for more steps...

Step 1.4.2.2.1

Simplify ${({(5 + t^{2})}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 1.4.2.2.1.1

Multiply the exponents in ${({(5 + t^{2})}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 1.4.2.2.1.1.1

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

${(5 + t^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.4.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.2.2.1.1.2.1

Cancel the common factor.

${(5 + t^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.4.2.2.1.1.2.2

Rewrite the expression.

${(5 + t^{2})}^{1} = 0^{2}$

Step 1.4.2.2.1.2

Simplify.

$5 + t^{2} = 0^{2}$

Step 1.4.2.3

Simplify the right side.

Tap for more steps...

Step 1.4.2.3.1

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

$5 + t^{2} = 0$

Step 1.4.3

Solve for $t$ .

Tap for more steps...

Step 1.4.3.1

Subtract $5$ from both sides of the equation.

$t^{2} = - 5$

Step 1.4.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{- 5}$

Step 1.4.3.3

Simplify $\pm \sqrt{- 5}$ .

Tap for more steps...

Step 1.4.3.3.1

Rewrite $- 5$ as $- 1 (5)$ .

$t = \pm \sqrt{- 1 (5)}$

Step 1.4.3.3.2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$t = \pm \sqrt{- 1} \cdot \sqrt{5}$

Step 1.4.3.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$t = \pm i \sqrt{5}$

Step 1.4.3.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 1.4.3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$t = i \sqrt{5}$

Step 1.4.3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$t = - i \sqrt{5}$

Step 1.4.3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$t = i \sqrt{5}, - i \sqrt{5}$

Step 1.5

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${t | t \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${t | t \in ℝ}$

Step 2

$g (t)$ is continuous on $[2, 5]$ .

$g (t)$ is continuous

Step 3

The average value of function $g$ over the interval $[a, b]$ is defined as $A (t) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (t) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Step 4

Substitute the actual values into the formula for the average value of a function.

$A (t) = \frac{1}{5 - 2} (\int_{2}^{5} \frac{t}{\sqrt{5 + t^{2}}} d t)$

Step 5

Let $u = 5 + t^{2}$ . Then $d u = 2 t d t$ , so $\frac{1}{2} d u = t d t$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 5.1

Let $u = 5 + t^{2}$ . Find $\frac{d u}{d t}$ .

Tap for more steps...

Step 5.1.1

Differentiate $5 + t^{2}$ .

$\frac{d}{d t} [5 + t^{2}]$

Step 5.1.2

By the Sum Rule, the derivative of $5 + t^{2}$ with respect to $t$ is $\frac{d}{d t} [5] + \frac{d}{d t} [t^{2}]$ .

$\frac{d}{d t} [5] + \frac{d}{d t} [t^{2}]$

Step 5.1.3

Since $5$ is constant with respect to $t$ , the derivative of $5$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [t^{2}]$

Step 5.1.4

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

$0 + 2 t$

Step 5.1.5

Add $0$ and $2 t$ .

$2 t$

Step 5.2

Substitute the lower limit in for $t$ in $u = 5 + t^{2}$ .

$u_{lower} = 5 + 2^{2}$

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

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

$u_{lower} = 5 + 4$

Step 5.3.2

Add $5$ and $4$ .

$u_{lower} = 9$

Step 5.4

Substitute the upper limit in for $t$ in $u = 5 + t^{2}$ .

$u_{upper} = 5 + 5^{2}$

Step 5.5

Simplify.

Tap for more steps...

Step 5.5.1

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

$u_{upper} = 5 + 25$

Step 5.5.2

Add $5$ and $25$ .

$u_{upper} = 30$

Step 5.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 9$

$u_{upper} = 30$

Step 5.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$A (t) = \frac{1}{5 - 2} (\int_{9}^{30} \frac{1}{\sqrt{u}} \cdot \frac{1}{2} d u)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{2}$ .

$A (t) = \frac{1}{5 - 2} (\int_{9}^{30} \frac{1}{\sqrt{u} \cdot 2} d u)$

Step 6.2

Move $2$ to the left of $\sqrt{u}$ .

$A (t) = \frac{1}{5 - 2} (\int_{9}^{30} \frac{1}{2 \sqrt{u}} d u)$

Step 7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot \int_{9}^{30} \frac{1}{\sqrt{u}} d u)$

Step 8

Apply basic rules of exponents.

Tap for more steps...

Step 8.1

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

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot \int_{9}^{30} \frac{1}{u^{\frac{1}{2}}} d u)$

Step 8.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot \int_{9}^{30} {(u^{\frac{1}{2}})}^{- 1} d u)$

Step 8.3

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

Tap for more steps...

Step 8.3.1

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

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot \int_{9}^{30} u^{\frac{1}{2} \cdot - 1} d u)$

Step 8.3.2

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

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot \int_{9}^{30} u^{\frac{- 1}{2}} d u)$

Step 8.3.3

Move the negative in front of the fraction.

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot \int_{9}^{30} u^{- \frac{1}{2}} d u)$

Step 9

By the Power Rule, the integral of $u^{- \frac{1}{2}}$ with respect to $u$ is $2 u^{\frac{1}{2}}$ .

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot 2 u^{\frac{1}{2}}]_{9}^{30})$

Step 10

Substitute and simplify.

Tap for more steps...

Step 10.1

Evaluate $2 u^{\frac{1}{2}}$ at $30$ and at $9$ .

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot ((2 \cdot 30^{\frac{1}{2}}) - 2 \cdot 9^{\frac{1}{2}}))$

Step 10.2

Simplify.

Tap for more steps...

Step 10.2.1

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

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot (2 \cdot 30^{\frac{1}{2}} - 2 \cdot {(3^{2})}^{\frac{1}{2}}))$

Step 10.2.2

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

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot (2 \cdot 30^{\frac{1}{2}} - 2 \cdot 3^{2 (\frac{1}{2})}))$

Step 10.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.2.3.1

Cancel the common factor.

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot (2 \cdot 30^{\frac{1}{2}} - 2 \cdot 3^{2 (\frac{1}{2})}))$

Step 10.2.3.2

Rewrite the expression.

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot (2 \cdot 30^{\frac{1}{2}} - 2 \cdot 3))$

Step 10.2.4

Evaluate the exponent.

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot (2 \cdot 30^{\frac{1}{2}} - 2 \cdot 3))$

Step 10.2.5

Multiply $- 2$ by $3$ .

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot (2 \cdot 30^{\frac{1}{2}} - 6))$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Apply the distributive property.

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot (2 \cdot 30^{\frac{1}{2}}) + \frac{1}{2} \cdot - 6)$

Step 11.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.2.1

Factor $2$ out of $2 \cdot 30^{\frac{1}{2}}$ .

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot (2 (30^{\frac{1}{2}})) + \frac{1}{2} \cdot - 6)$

Step 11.2.2

Cancel the common factor.

$A (t) = \frac{1}{5 - 2} (\frac{1}{2} \cdot (2 \cdot 30^{\frac{1}{2}}) + \frac{1}{2} \cdot - 6)$

Step 11.2.3

Rewrite the expression.

$A (t) = \frac{1}{5 - 2} (30^{\frac{1}{2}} + \frac{1}{2} \cdot - 6)$

Step 11.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.3.1

Factor $2$ out of $- 6$ .

$A (t) = \frac{1}{5 - 2} (30^{\frac{1}{2}} + \frac{1}{2} \cdot (2 (- 3)))$

Step 11.3.2

Cancel the common factor.

$A (t) = \frac{1}{5 - 2} (30^{\frac{1}{2}} + \frac{1}{2} \cdot (2 \cdot - 3))$

Step 11.3.3

Rewrite the expression.

$A (t) = \frac{1}{5 - 2} (30^{\frac{1}{2}} - 3)$

Step 12

Subtract $2$ from $5$ .

$A (t) = \frac{1}{3} \cdot (30^{\frac{1}{2}} - 3)$

Step 13

Apply the distributive property.

$A (t) = \frac{1}{3} \cdot 30^{\frac{1}{2}} + \frac{1}{3} \cdot - 3$

Step 14

Combine $\frac{1}{3}$ and $30^{\frac{1}{2}}$ .

$A (t) = \frac{30^{\frac{1}{2}}}{3} + \frac{1}{3} \cdot - 3$

Step 15

Cancel the common factor of $3$ .

Tap for more steps...

Step 15.1

Factor $3$ out of $- 3$ .

$A (t) = \frac{30^{\frac{1}{2}}}{3} + \frac{1}{3} \cdot (3 (- 1))$

Step 15.2

Cancel the common factor.

$A (t) = \frac{30^{\frac{1}{2}}}{3} + \frac{1}{3} \cdot (3 \cdot - 1)$

Step 15.3

Rewrite the expression.

$A (t) = \frac{30^{\frac{1}{2}}}{3} - 1$

Step 16