Calculus Examples

$- 3 t^{2} + t + 70$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $- 3 t^{2} + t + 70$ with respect to $t$ is $\frac{d}{d t} [- 3 t^{2}] + \frac{d}{d t} [t] + \frac{d}{d t} [70]$ .

$\frac{d}{d t} [- 3 t^{2}] + \frac{d}{d t} [t] + \frac{d}{d t} [70]$

Step 1.1.2

Evaluate $\frac{d}{d t} [- 3 t^{2}]$ .

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Step 1.1.2.1

Since $- 3$ is constant with respect to $t$ , the derivative of $- 3 t^{2}$ with respect to $t$ is $- 3 \frac{d}{d t} [t^{2}]$ .

$- 3 \frac{d}{d t} [t^{2}] + \frac{d}{d t} [t] + \frac{d}{d t} [70]$

Step 1.1.2.2

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

$- 3 (2 t) + \frac{d}{d t} [t] + \frac{d}{d t} [70]$

Step 1.1.2.3

Multiply $2$ by $- 3$ .

$- 6 t + \frac{d}{d t} [t] + \frac{d}{d t} [70]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

$- 6 t + 1 + \frac{d}{d t} [70]$

Step 1.1.3.2

Since $70$ is constant with respect to $t$ , the derivative of $70$ with respect to $t$ is $0$ .

$- 6 t + 1 + 0$

Step 1.1.3.3

Add $- 6 t + 1$ and $0$ .

$f' (t) = - 6 t + 1$

Step 1.2

The first derivative of $f (t)$ with respect to $t$ is $- 6 t + 1$ .

$- 6 t + 1$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 6 t + 1 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 6 t + 1 = 0$

Step 2.2

Subtract $1$ from both sides of the equation.

$- 6 t = - 1$

Step 2.3

Divide each term in $- 6 t = - 1$ by $- 6$ and simplify.

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Step 2.3.1

Divide each term in $- 6 t = - 1$ by $- 6$ .

$\frac{- 6 t}{- 6} = \frac{- 1}{- 6}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 6$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 6 t}{- 6} = \frac{- 1}{- 6}$

Step 2.3.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 1}{- 6}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Dividing two negative values results in a positive value.

$t = \frac{1}{6}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $- 3 t^{2} + t + 70$ at each $t$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $t = \frac{1}{6}$ .

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Step 4.1.1

Substitute $\frac{1}{6}$ for $t$ .

$- 3 {(\frac{1}{6})}^{2} + \frac{1}{6} + 70$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Remove parentheses.

$- 3 {(\frac{1}{6})}^{2} + \frac{1}{6} + 70$

Step 4.1.2.2

Simplify each term.

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Step 4.1.2.2.1

Apply the product rule to $\frac{1}{6}$ .

$- 3 \frac{1^{2}}{6^{2}} + \frac{1}{6} + 70$

Step 4.1.2.2.2

One to any power is one.

$- 3 \frac{1}{6^{2}} + \frac{1}{6} + 70$

Step 4.1.2.2.3

Raise $6$ to the power of $2$ .

$- 3 (\frac{1}{36}) + \frac{1}{6} + 70$

Step 4.1.2.2.4

Cancel the common factor of $3$ .

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Step 4.1.2.2.4.1

Factor $3$ out of $- 3$ .

$3 (- 1) \frac{1}{36} + \frac{1}{6} + 70$

Step 4.1.2.2.4.2

Factor $3$ out of $36$ .

$3 \cdot - 1 \frac{1}{3 \cdot 12} + \frac{1}{6} + 70$

Step 4.1.2.2.4.3

Cancel the common factor.

$3 \cdot - 1 \frac{1}{3 \cdot 12} + \frac{1}{6} + 70$

Step 4.1.2.2.4.4

Rewrite the expression.

$- 1 (\frac{1}{12}) + \frac{1}{6} + 70$

Step 4.1.2.2.5

Rewrite $- 1 (\frac{1}{12})$ as $- (\frac{1}{12})$ .

$- \frac{1}{12} + \frac{1}{6} + 70$

Step 4.1.2.3

Find the common denominator.

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Step 4.1.2.3.1

Multiply $\frac{1}{6}$ by $\frac{2}{2}$ .

$- \frac{1}{12} + \frac{1}{6} \cdot \frac{2}{2} + 70$

Step 4.1.2.3.2

Multiply $\frac{1}{6}$ by $\frac{2}{2}$ .

$- \frac{1}{12} + \frac{2}{6 \cdot 2} + 70$

Step 4.1.2.3.3

Write $70$ as a fraction with denominator $1$ .

$- \frac{1}{12} + \frac{2}{6 \cdot 2} + \frac{70}{1}$

Step 4.1.2.3.4

Multiply $\frac{70}{1}$ by $\frac{12}{12}$ .

$- \frac{1}{12} + \frac{2}{6 \cdot 2} + \frac{70}{1} \cdot \frac{12}{12}$

Step 4.1.2.3.5

Multiply $\frac{70}{1}$ by $\frac{12}{12}$ .

$- \frac{1}{12} + \frac{2}{6 \cdot 2} + \frac{70 \cdot 12}{12}$

Step 4.1.2.3.6

Reorder the factors of $6 \cdot 2$ .

$- \frac{1}{12} + \frac{2}{2 \cdot 6} + \frac{70 \cdot 12}{12}$

Step 4.1.2.3.7

Multiply $2$ by $6$ .

$- \frac{1}{12} + \frac{2}{12} + \frac{70 \cdot 12}{12}$

Step 4.1.2.4

Combine the numerators over the common denominator.

$\frac{- 1 + 2 + 70 \cdot 12}{12}$

Step 4.1.2.5

Simplify the expression.

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Step 4.1.2.5.1

Multiply $70$ by $12$ .

$\frac{- 1 + 2 + 840}{12}$

Step 4.1.2.5.2

Add $- 1$ and $2$ .

$\frac{1 + 840}{12}$

Step 4.1.2.5.3

Add $1$ and $840$ .

$\frac{841}{12}$

Step 4.2

List all of the points.

$(\frac{1}{6}, \frac{841}{12})$

Step 5