Calculus Examples

$y = 3 x^{2} + 3 x$ , $(1, 6)$

Step 1

Write $y = 3 x^{2} + 3 x$ as a function.

$f (x) = 3 x^{2} + 3 x$

Step 2

Check if the given point is on the graph of the given function.

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Step 2.1

Evaluate $f (x) = 3 x^{2} + 3 x$ at $x = 1$ .

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Step 2.1.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 3 {(1)}^{2} + 3 (1)$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

One to any power is one.

$f (1) = 3 \cdot 1 + 3 (1)$

Step 2.1.2.1.2

Multiply $3$ by $1$ .

$f (1) = 3 + 3 (1)$

Step 2.1.2.1.3

Multiply $3$ by $1$ .

$f (1) = 3 + 3$

Step 2.1.2.2

Add $3$ and $3$ .

$f (1) = 6$

Step 2.1.2.3

The final answer is $6$ .

$6$

Step 2.2

Since $6 = 6$ , the point is on the graph.

The point is on the graph

Step 3

The slope of the tangent line is the derivative of the expression.

$m$ $=$ The derivative of $f (x) = 3 x^{2} + 3 x$

Step 4

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 5

Find the components of the definition.

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Step 5.1

Evaluate the function at $x = x + h$ .

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Step 5.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 3 {(x + h)}^{2} + 3 (x + h)$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = 3 ((x + h) (x + h)) + 3 (x + h)$

Step 5.1.2.1.2

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 5.1.2.1.2.1

Apply the distributive property.

$f (x + h) = 3 (x (x + h) + h (x + h)) + 3 (x + h)$

Step 5.1.2.1.2.2

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h (x + h)) + 3 (x + h)$

Step 5.1.2.1.2.3

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h x + h \cdot h) + 3 (x + h)$

Step 5.1.2.1.3

Simplify and combine like terms.

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Step 5.1.2.1.3.1

Simplify each term.

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Step 5.1.2.1.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = 3 (x^{2} + x h + h x + h \cdot h) + 3 (x + h)$

Step 5.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = 3 (x^{2} + x h + h x + h^{2}) + 3 (x + h)$

Step 5.1.2.1.3.2

Add $x h$ and $h x$ .

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Step 5.1.2.1.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = 3 (x^{2} + h x + h x + h^{2}) + 3 (x + h)$

Step 5.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = 3 (x^{2} + 2 h x + h^{2}) + 3 (x + h)$

Step 5.1.2.1.4

Apply the distributive property.

$f (x + h) = 3 x^{2} + 3 (2 h x) + 3 h^{2} + 3 (x + h)$

Step 5.1.2.1.5

Multiply $2$ by $3$ .

$f (x + h) = 3 x^{2} + 6 (h x) + 3 h^{2} + 3 (x + h)$

Step 5.1.2.1.6

Apply the distributive property.

$f (x + h) = 3 x^{2} + 6 h x + 3 h^{2} + 3 x + 3 h$

Step 5.1.2.2

The final answer is $3 x^{2} + 6 h x + 3 h^{2} + 3 x + 3 h$ .

$3 x^{2} + 6 h x + 3 h^{2} + 3 x + 3 h$

Step 5.2

Reorder.

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Step 5.2.1

Move $3 x$ .

$3 x^{2} + 6 h x + 3 h^{2} + 3 h + 3 x$

Step 5.2.2

Move $3 x^{2}$ .

$6 h x + 3 h^{2} + 3 x^{2} + 3 h + 3 x$

Step 5.2.3

Reorder $6 h x$ and $3 h^{2}$ .

$3 h^{2} + 6 h x + 3 x^{2} + 3 h + 3 x$

Step 5.3

Find the components of the definition.

$f (x + h) = 3 h^{2} + 6 h x + 3 x^{2} + 3 h + 3 x$

$f (x) = 3 x^{2} + 3 x$

$f (x + h) = 3 h^{2} + 6 h x + 3 x^{2} + 3 h + 3 x$

$f (x) = 3 x^{2} + 3 x$

Step 6

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{3 h^{2} + 6 h x + 3 x^{2} + 3 h + 3 x - (3 x^{2} + 3 x)}{h}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{3 h^{2} + 6 h x + 3 x^{2} + 3 h + 3 x - (3 x^{2}) - (3 x)}{h}$

Step 7.1.2

Multiply $3$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{2} + 6 h x + 3 x^{2} + 3 h + 3 x - 3 x^{2} - (3 x)}{h}$

Step 7.1.3

Multiply $3$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{2} + 6 h x + 3 x^{2} + 3 h + 3 x - 3 x^{2} - 3 x}{h}$

Step 7.1.4

Subtract $3 x^{2}$ from $3 x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{2} + 6 h x + 3 h + 3 x + 0 - 3 x}{h}$

Step 7.1.5

Add $3 h^{2}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{2} + 6 h x + 3 h + 3 x - 3 x}{h}$

Step 7.1.6

Subtract $3 x$ from $3 x$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{2} + 6 h x + 3 h + 0}{h}$

Step 7.1.7

Add $3 h^{2} + 6 h x + 3 h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{3 h^{2} + 6 h x + 3 h}{h}$

Step 7.1.8

Factor $3 h$ out of $3 h^{2} + 6 h x + 3 h$ .

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Step 7.1.8.1

Factor $3 h$ out of $3 h^{2}$ .

$f' (x) = lim_{h \to 0} \frac{3 h \cdot h + 6 h x + 3 h}{h}$

Step 7.1.8.2

Factor $3 h$ out of $6 h x$ .

$f' (x) = lim_{h \to 0} \frac{3 h \cdot h + 3 h (2 x) + 3 h}{h}$

Step 7.1.8.3

Factor $3 h$ out of $3 h$ .

$f' (x) = lim_{h \to 0} \frac{3 h \cdot h + 3 h (2 x) + 3 h \cdot 1}{h}$

Step 7.1.8.4

Factor $3 h$ out of $3 h \cdot h + 3 h (2 x)$ .

$f' (x) = lim_{h \to 0} \frac{3 h (h + 2 x) + 3 h \cdot 1}{h}$

Step 7.1.8.5

Factor $3 h$ out of $3 h (h + 2 x) + 3 h \cdot 1$ .

$f' (x) = lim_{h \to 0} \frac{3 h (h + 2 x + 1)}{h}$

Step 7.2

Simplify terms.

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Step 7.2.1

Cancel the common factor of $h$ .

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Step 7.2.1.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{3 h (h + 2 x + 1)}{h}$

Step 7.2.1.2

Divide $3 (h + 2 x + 1)$ by $1$ .

$f' (x) = lim_{h \to 0} 3 (h + 2 x + 1)$

Step 7.2.2

Apply the distributive property.

$f' (x) = lim_{h \to 0} 3 h + 3 (2 x) + 3 \cdot 1$

Step 7.3

Simplify.

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Step 7.3.1

Multiply $2$ by $3$ .

$f' (x) = lim_{h \to 0} 3 h + 6 x + 3 \cdot 1$

Step 7.3.2

Multiply $3$ by $1$ .

$f' (x) = lim_{h \to 0} 3 h + 6 x + 3$

Step 7.4

Reorder $3 h$ and $6 x$ .

$f' (x) = lim_{h \to 0} 6 x + 3 h + 3$

Step 8

Evaluate the limit.

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Step 8.1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$lim_{h \to 0} 6 x + lim_{h \to 0} 3 h + lim_{h \to 0} 3$

Step 8.2

Evaluate the limit of $6 x$ which is constant as $h$ approaches $0$ .

$6 x + lim_{h \to 0} 3 h + lim_{h \to 0} 3$

Step 8.3

Move the term $3$ outside of the limit because it is constant with respect to $h$ .

$6 x + 3 lim_{h \to 0} h + lim_{h \to 0} 3$

Step 8.4

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

$6 x + 3 lim_{h \to 0} h + 3$

Step 9

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$6 x + 3 \cdot 0 + 3$

Step 10

Simplify the answer.

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Step 10.1

Multiply $3$ by $0$ .

$6 x + 0 + 3$

Step 10.2

Add $6 x$ and $0$ .

$6 x + 3$

Step 11

Simplify $6 (1) + 3$ .

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Step 11.1

Multiply $6$ by $1$ .

$m = 6 + 3$

Step 11.2

Add $6$ and $3$ .

$m = 9$

Step 12

The slope is $m = 9$ and the point is $(1, 6)$ .

$m = 9, (1, 6)$

Step 13

Find the value of $b$ using the formula for the equation of a line.

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Step 13.1

Use the formula for the equation of a line to find $b$ .

$y = m x + b$

Step 13.2

Substitute the value of $m$ into the equation.

$y = (9) \cdot x + b$

Step 13.3

Substitute the value of $x$ into the equation.

$y = (9) \cdot (1) + b$

Step 13.4

Substitute the value of $y$ into the equation.

$6 = (9) \cdot (1) + b$

Step 13.5

Find the value of $b$ .

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Step 13.5.1

Rewrite the equation as $(9) \cdot (1) + b = 6$ .

$(9) \cdot (1) + b = 6$

Step 13.5.2

Multiply $9$ by $1$ .

$9 + b = 6$

Step 13.5.3

Move all terms not containing $b$ to the right side of the equation.

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Step 13.5.3.1

Subtract $9$ from both sides of the equation.

$b = 6 - 9$

Step 13.5.3.2

Subtract $9$ from $6$ .

$b = - 3$

Step 14

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

$y = 9 x - 3$

Step 15

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