Calculus Examples

$f (3) = \frac{5}{6} t^{3} - \frac{4}{9} t^{4} + 25 e^{- t} + 45 t + 4$

Step 1

Replace the variable $t$ with $3$ in the expression.

$f (3) = \frac{5}{6} \cdot {(3)}^{3} - \frac{4}{9} \cdot {(3)}^{4} + 25 e^{- (3)} + 45 (3) + 4$

Step 2

Simplify the result.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $3$ to the power of $3$ .

$f (3) = \frac{5}{6} \cdot 27 - \frac{4}{9} \cdot {(3)}^{4} + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.2

Cancel the common factor of $3$ .

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Step 2.1.2.1

Factor $3$ out of $6$ .

$f (3) = \frac{5}{3 (2)} \cdot 27 - \frac{4}{9} \cdot {(3)}^{4} + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.2.2

Factor $3$ out of $27$ .

$f (3) = \frac{5}{3 \cdot 2} \cdot (3 \cdot 9) - \frac{4}{9} \cdot {(3)}^{4} + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.2.3

Cancel the common factor.

$f (3) = \frac{5}{3 \cdot 2} \cdot (3 \cdot 9) - \frac{4}{9} \cdot {(3)}^{4} + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.2.4

Rewrite the expression.

$f (3) = \frac{5}{2} \cdot 9 - \frac{4}{9} \cdot {(3)}^{4} + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.3

Combine $\frac{5}{2}$ and $9$ .

$f (3) = \frac{5 \cdot 9}{2} - \frac{4}{9} \cdot {(3)}^{4} + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.4

Multiply $5$ by $9$ .

$f (3) = \frac{45}{2} - \frac{4}{9} \cdot {(3)}^{4} + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.5

Raise $3$ to the power of $4$ .

$f (3) = \frac{45}{2} - \frac{4}{9} \cdot 81 + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.6

Cancel the common factor of $9$ .

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Step 2.1.6.1

Move the leading negative in $- \frac{4}{9}$ into the numerator.

$f (3) = \frac{45}{2} + \frac{- 4}{9} \cdot 81 + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.6.2

Factor $9$ out of $81$ .

$f (3) = \frac{45}{2} + \frac{- 4}{9} \cdot (9 (9)) + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.6.3

Cancel the common factor.

$f (3) = \frac{45}{2} + \frac{- 4}{9} \cdot (9 \cdot 9) + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.6.4

Rewrite the expression.

$f (3) = \frac{45}{2} - 4 \cdot 9 + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.7

Multiply $- 4$ by $9$ .

$f (3) = \frac{45}{2} - 36 + 25 e^{- (3)} + 45 (3) + 4$

Step 2.1.8

Multiply $- 1$ by $3$ .

$f (3) = \frac{45}{2} - 36 + 25 e^{- 3} + 45 (3) + 4$

Step 2.1.9

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

$f (3) = \frac{45}{2} - 36 + 25 (\frac{1}{e^{3}}) + 45 (3) + 4$

Step 2.1.10

Combine $25$ and $\frac{1}{e^{3}}$ .

$f (3) = \frac{45}{2} - 36 + \frac{25}{e^{3}} + 45 (3) + 4$

Step 2.1.11

Multiply $45$ by $3$ .

$f (3) = \frac{45}{2} - 36 + \frac{25}{e^{3}} + 135 + 4$

Step 2.2

To write $- 36$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (3) = \frac{45}{2} - 36 \cdot \frac{2}{2} + \frac{25}{e^{3}} + 135 + 4$

Step 2.3

Combine $- 36$ and $\frac{2}{2}$ .

$f (3) = \frac{45}{2} + \frac{- 36 \cdot 2}{2} + \frac{25}{e^{3}} + 135 + 4$

Step 2.4

Combine the numerators over the common denominator.

$f (3) = \frac{45 - 36 \cdot 2}{2} + \frac{25}{e^{3}} + 135 + 4$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 36$ by $2$ .

$f (3) = \frac{45 - 72}{2} + \frac{25}{e^{3}} + 135 + 4$

Step 2.5.2

Subtract $72$ from $45$ .

$f (3) = \frac{- 27}{2} + \frac{25}{e^{3}} + 135 + 4$

Step 2.6

Move the negative in front of the fraction.

$f (3) = - \frac{27}{2} + \frac{25}{e^{3}} + 135 + 4$

Step 2.7

To write $135$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (3) = \frac{25}{e^{3}} - \frac{27}{2} + 135 \cdot \frac{2}{2} + 4$

Step 2.8

Combine $135$ and $\frac{2}{2}$ .

$f (3) = \frac{25}{e^{3}} - \frac{27}{2} + \frac{135 \cdot 2}{2} + 4$

Step 2.9

Combine the numerators over the common denominator.

$f (3) = \frac{25}{e^{3}} + \frac{- 27 + 135 \cdot 2}{2} + 4$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $135$ by $2$ .

$f (3) = \frac{25}{e^{3}} + \frac{- 27 + 270}{2} + 4$

Step 2.10.2

Add $- 27$ and $270$ .

$f (3) = \frac{25}{e^{3}} + \frac{243}{2} + 4$

Step 2.11

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (3) = \frac{25}{e^{3}} + \frac{243}{2} + 4 \cdot \frac{2}{2}$

Step 2.12

Combine $4$ and $\frac{2}{2}$ .

$f (3) = \frac{25}{e^{3}} + \frac{243}{2} + \frac{4 \cdot 2}{2}$

Step 2.13

Combine the numerators over the common denominator.

$f (3) = \frac{25}{e^{3}} + \frac{243 + 4 \cdot 2}{2}$

Step 2.14

Simplify the numerator.

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Step 2.14.1

Multiply $4$ by $2$ .

$f (3) = \frac{25}{e^{3}} + \frac{243 + 8}{2}$

Step 2.14.2

Add $243$ and $8$ .

$f (3) = \frac{25}{e^{3}} + \frac{251}{2}$

Step 2.15

The final answer is $\frac{25}{e^{3}} + \frac{251}{2}$ .

$\frac{25}{e^{3}} + \frac{251}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{25}{e^{3}} + \frac{251}{2}$

Decimal Form:

$126.74467670 \dots$

Step 4