Calculus Examples

$f (x) = x^{2} - 5 x + 6$ , $(1, 2)$

Step 1

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

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Step 1.1

Evaluate $f (x) = x^{2} - 5 x + 6$ at $x = 1$ .

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Step 1.1.1

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

$f (1) = {(1)}^{2} - 5 \cdot 1 + 6$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

One to any power is one.

$f (1) = 1 - 5 \cdot 1 + 6$

Step 1.1.2.1.2

Multiply $- 5$ by $1$ .

$f (1) = 1 - 5 + 6$

Step 1.1.2.2

Simplify by adding and subtracting.

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Step 1.1.2.2.1

Subtract $5$ from $1$ .

$f (1) = - 4 + 6$

Step 1.1.2.2.2

Add $- 4$ and $6$ .

$f (1) = 2$

Step 1.1.2.3

The final answer is $2$ .

$2$

Step 1.2

Since $2 = 2$ , the point is on the graph.

The point is on the graph

Step 2

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

$m$ $=$ The derivative of $f (x) = x^{2} - 5 x + 6$

Step 3

Consider the limit definition of the derivative.

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

Step 4

Find the components of the definition.

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Step 4.1

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

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Step 4.1.1

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

$f (x + h) = {(x + h)}^{2} - 5 (x + h) + 6$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$f (x + h) = (x + h) (x + h) - 5 (x + h) + 6$

Step 4.1.2.1.2

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

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Step 4.1.2.1.2.1

Apply the distributive property.

$f (x + h) = x (x + h) + h (x + h) - 5 (x + h) + 6$

Step 4.1.2.1.2.2

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h (x + h) - 5 (x + h) + 6$

Step 4.1.2.1.2.3

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h x + h \cdot h - 5 (x + h) + 6$

Step 4.1.2.1.3

Simplify and combine like terms.

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Step 4.1.2.1.3.1

Simplify each term.

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Step 4.1.2.1.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = x^{2} + x h + h x + h \cdot h - 5 (x + h) + 6$

Step 4.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = x^{2} + x h + h x + h^{2} - 5 (x + h) + 6$

Step 4.1.2.1.3.2

Add $x h$ and $h x$ .

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Step 4.1.2.1.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = x^{2} + h x + h x + h^{2} - 5 (x + h) + 6$

Step 4.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = x^{2} + 2 h x + h^{2} - 5 (x + h) + 6$

Step 4.1.2.1.4

Apply the distributive property.

$f (x + h) = x^{2} + 2 h x + h^{2} - 5 x - 5 h + 6$

Step 4.1.2.2

The final answer is $x^{2} + 2 h x + h^{2} - 5 x - 5 h + 6$ .

$x^{2} + 2 h x + h^{2} - 5 x - 5 h + 6$

Step 4.2

Reorder.

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Step 4.2.1

Move $- 5 x$ .

$x^{2} + 2 h x + h^{2} - 5 h - 5 x + 6$

Step 4.2.2

Move $x^{2}$ .

$2 h x + h^{2} + x^{2} - 5 h - 5 x + 6$

Step 4.2.3

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

$h^{2} + 2 h x + x^{2} - 5 h - 5 x + 6$

Step 4.3

Find the components of the definition.

$f (x + h) = h^{2} + 2 h x + x^{2} - 5 h - 5 x + 6$

$f (x) = x^{2} - 5 x + 6$

$f (x + h) = h^{2} + 2 h x + x^{2} - 5 h - 5 x + 6$

$f (x) = x^{2} - 5 x + 6$

Step 5

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} - 5 h - 5 x + 6 - (x^{2} - 5 x + 6)}{h}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} - 5 h - 5 x + 6 - x^{2} - (- 5 x) - 1 \cdot 6}{h}$

Step 6.1.2

Simplify.

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Step 6.1.2.1

Multiply $- 5$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} - 5 h - 5 x + 6 - x^{2} + 5 x - 1 \cdot 6}{h}$

Step 6.1.2.2

Multiply $- 1$ by $6$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x + x^{2} - 5 h - 5 x + 6 - x^{2} + 5 x - 6}{h}$

Step 6.1.3

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

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x - 5 h - 5 x + 6 + 0 + 5 x - 6}{h}$

Step 6.1.4

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

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x - 5 h - 5 x + 6 + 5 x - 6}{h}$

Step 6.1.5

Add $- 5 x$ and $5 x$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x - 5 h + 0 + 6 - 6}{h}$

Step 6.1.6

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

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x - 5 h + 6 - 6}{h}$

Step 6.1.7

Subtract $6$ from $6$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x - 5 h + 0}{h}$

Step 6.1.8

Add $h^{2} + 2 h x - 5 h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 2 h x - 5 h}{h}$

Step 6.1.9

Factor $h$ out of $h^{2} + 2 h x - 5 h$ .

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Step 6.1.9.1

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

$f' (x) = lim_{h \to 0} \frac{h \cdot h + 2 h x - 5 h}{h}$

Step 6.1.9.2

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

$f' (x) = lim_{h \to 0} \frac{h (h) + h (2 x) - 5 h}{h}$

Step 6.1.9.3

Factor $h$ out of $- 5 h$ .

$f' (x) = lim_{h \to 0} \frac{h (h) + h (2 x) + h \cdot - 5}{h}$

Step 6.1.9.4

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

$f' (x) = lim_{h \to 0} \frac{h (h + 2 x) + h \cdot - 5}{h}$

Step 6.1.9.5

Factor $h$ out of $h (h + 2 x) + h \cdot - 5$ .

$f' (x) = lim_{h \to 0} \frac{h (h + 2 x - 5)}{h}$

Step 6.2

Reduce the expression by cancelling the common factors.

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Step 6.2.1

Cancel the common factor of $h$ .

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Step 6.2.1.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h (h + 2 x - 5)}{h}$

Step 6.2.1.2

Divide $h + 2 x - 5$ by $1$ .

$f' (x) = lim_{h \to 0} h + 2 x - 5$

Step 6.2.2

Reorder $h$ and $2 x$ .

$f' (x) = lim_{h \to 0} 2 x + h - 5$

Step 7

Evaluate the limit.

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Step 7.1

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

$lim_{h \to 0} 2 x + lim_{h \to 0} h - lim_{h \to 0} 5$

Step 7.2

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

$2 x + lim_{h \to 0} h - lim_{h \to 0} 5$

Step 7.3

Evaluate the limit of $5$ which is constant as $h$ approaches $0$ .

$2 x + lim_{h \to 0} h - 1 \cdot 5$

Step 8

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

$2 x + 0 - 1 \cdot 5$

Step 9

Simplify the answer.

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Step 9.1

Add $2 x$ and $0$ .

$2 x - 1 \cdot 5$

Step 9.2

Multiply $- 1$ by $5$ .

$2 x - 5$

Step 10

Simplify $2 (1) - 5$ .

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Step 10.1

Multiply $2$ by $1$ .

$m = 2 - 5$

Step 10.2

Subtract $5$ from $2$ .

$m = - 3$

Step 11

The slope is $m = - 3$ and the point is $(1, 2)$ .

$m = - 3, (1, 2)$

Step 12

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

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Step 12.1

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

$y = m x + b$

Step 12.2

Substitute the value of $m$ into the equation.

$y = (- 3) \cdot x + b$

Step 12.3

Substitute the value of $x$ into the equation.

$y = (- 3) \cdot (1) + b$

Step 12.4

Substitute the value of $y$ into the equation.

$2 = (- 3) \cdot (1) + b$

Step 12.5

Find the value of $b$ .

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Step 12.5.1

Rewrite the equation as $(- 3) \cdot (1) + b = 2$ .

$(- 3) \cdot (1) + b = 2$

Step 12.5.2

Multiply $- 3$ by $1$ .

$- 3 + b = 2$

Step 12.5.3

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

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Step 12.5.3.1

Add $3$ to both sides of the equation.

$b = 2 + 3$

Step 12.5.3.2

Add $2$ and $3$ .

$b = 5$

Step 13

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.

$y = - 3 x + 5$

Step 14