Calculus Examples

$y = {(x + 3)}^{2}$ , $(1, 16)$

Step 1

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

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Add $1$ and $3$ .

$f (1) = 4^{2}$

Step 2.1.2.2

Raise $4$ to the power of $2$ .

$f (1) = 16$

Step 2.1.2.3

The final answer is $16$ .

$16$

Step 2.2

Since $16 = 16$ , 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 $f (x) = {(x + 3)}^{2}$

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.

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

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Rewrite ${(x + h + 3)}^{2}$ as $(x + h + 3) (x + h + 3)$ .

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

Step 5.1.2.2

Expand $(x + h + 3) (x + h + 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 5.1.2.3

Simplify each term.

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Step 5.1.2.3.1

Multiply $x$ by $x$ .

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

Step 5.1.2.3.2

Move $3$ to the left of $x$ .

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

Step 5.1.2.3.3

Multiply $h$ by $h$ .

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

Step 5.1.2.3.4

Move $3$ to the left of $h$ .

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

Step 5.1.2.3.5

Multiply $3$ by $3$ .

$f (x + h) = x^{2} + x h + 3 x + h x + h^{2} + 3 h + 3 x + 3 h + 9$

Step 5.1.2.4

Add $x h$ and $h x$ .

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Step 5.1.2.4.1

Reorder $x$ and $h$ .

$f (x + h) = x^{2} + h x + h x + 3 x + h^{2} + 3 h + 3 x + 3 h + 9$

Step 5.1.2.4.2

Add $h x$ and $h x$ .

$f (x + h) = x^{2} + 2 h x + 3 x + h^{2} + 3 h + 3 x + 3 h + 9$

Step 5.1.2.5

Add $3 x$ and $3 x$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 3 h + 6 x + 3 h + 9$

Step 5.1.2.6

Add $3 h$ and $3 h$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 6 h + 6 x + 9$

Step 5.1.2.7

The final answer is $x^{2} + 2 h x + h^{2} + 6 h + 6 x + 9$ .

$x^{2} + 2 h x + h^{2} + 6 h + 6 x + 9$

Step 5.2

Reorder.

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Step 5.2.1

Move $x^{2}$ .

$2 h x + h^{2} + x^{2} + 6 h + 6 x + 9$

Step 5.2.2

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

$h^{2} + 2 h x + x^{2} + 6 h + 6 x + 9$

Step 5.3

Find the components of the definition.

$f (x + h) = h^{2} + 2 h x + x^{2} + 6 h + 6 x + 9$

$f (x) = x^{2} + 6 x + 9$

$f (x + h) = h^{2} + 2 h x + x^{2} + 6 h + 6 x + 9$

$f (x) = x^{2} + 6 x + 9$

Step 6

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} + 6 h + 6 x + 9 - (x^{2} + 6 x + 9)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} + 6 h + 6 x + 9 - x^{2} - (6 x) - 1 \cdot 9}{h}$

Step 7.1.2

Simplify.

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Step 7.1.2.1

Multiply $6$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} + 6 h + 6 x + 9 - x^{2} - 6 x - 1 \cdot 9}{h}$

Step 7.1.2.2

Multiply $- 1$ by $9$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} + 6 h + 6 x + 9 - x^{2} - 6 x - 9}{h}$

Step 7.1.3

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

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + 6 h + 6 x + 9 + 0 - 6 x - 9}{h}$

Step 7.1.4

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

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + 6 h + 6 x + 9 - 6 x - 9}{h}$

Step 7.1.5

Subtract $6 x$ from $6 x$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + 6 h + 0 + 9 - 9}{h}$

Step 7.1.6

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

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + 6 h + 9 - 9}{h}$

Step 7.1.7

Subtract $9$ from $9$ .

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

Step 7.1.8

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

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

Step 7.1.9

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

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Step 7.1.9.1

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

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

Step 7.1.9.2

Factor $h$ out of $2 h x$ .

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

Step 7.1.9.3

Factor $h$ out of $6 h$ .

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

Step 7.1.9.4

Factor $h$ out of $h (h) + h (2 x)$ .

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

Step 7.1.9.5

Factor $h$ out of $h (h + 2 x) + h \cdot 6$ .

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

Step 7.2

Reduce the expression by cancelling the common factors.

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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.

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

Step 7.2.1.2

Divide $h + 2 x + 6$ by $1$ .

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

Step 7.2.2

Reorder $h$ and $2 x$ .

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

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 $0$ .

$lim_{h \to 0} 2 x + lim_{h \to 0} h + lim_{h \to 0} 6$

Step 8.2

Evaluate the limit of $2 x$ which is constant as $h$ approaches $0$ .

$2 x + lim_{h \to 0} h + lim_{h \to 0} 6$

Step 8.3

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

$2 x + lim_{h \to 0} h + 6$

Step 9

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

$2 x + 0 + 6$

Step 10

Add $2 x$ and $0$ .

$2 x + 6$

Step 11

Simplify $2 (1) + 6$ .

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Step 11.1

Multiply $2$ by $1$ .

$m = 2 + 6$

Step 11.2

Add $2$ and $6$ .

$m = 8$

Step 12

The slope is $m = 8$ and the point is $(1, 16)$ .

$m = 8, (1, 16)$

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.

$y = (8) \cdot x + b$

Step 13.3

Substitute the value of $x$ into the equation.

$y = (8) \cdot (1) + b$

Step 13.4

Substitute the value of $y$ into the equation.

$16 = (8) \cdot (1) + b$

Step 13.5

Find the value of $b$ .

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Step 13.5.1

Rewrite the equation as $(8) \cdot (1) + b = 16$ .

$(8) \cdot (1) + b = 16$

Step 13.5.2

Multiply $8$ by $1$ .

$8 + b = 16$

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 $8$ from both sides of the equation.

$b = 16 - 8$

Step 13.5.3.2

Subtract $8$ from $16$ .

$b = 8$

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.

$y = 8 x + 8$

Step 15

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