Calculus Examples

$\sqrt{x}$ , $(1, 1)$

Step 1

Write $\sqrt{x}$ as a function.

$f (x) = \sqrt{x}$

Step 2

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

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Step 2.1

Evaluate $f (x) = \sqrt{x}$ at $x = 1$ .

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Step 2.1.1

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

$f (1) = \sqrt{1}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (1) = \sqrt{1}$

Step 2.1.2.2

Any root of $1$ is $1$ .

$f (1) = 1$

Step 2.1.2.3

The final answer is $1$ .

$1$

Step 2.2

Since $1 = 1$ , 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 $f (x) = \sqrt{x}$

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.

$f (x + h) = \sqrt{x + h}$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Remove parentheses.

$f (x + h) = \sqrt{x + h}$

Step 5.1.2.2

The final answer is $\sqrt{x + h}$ .

$\sqrt{x + h}$

Step 5.2

Find the components of the definition.

$f (x + h) = \sqrt{x + h}$

$f (x) = \sqrt{x}$

$f (x + h) = \sqrt{x + h}$

$f (x) = \sqrt{x}$

Step 6

Plug in the components.

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

Step 7

Multiply $- 1$ by $\sqrt{x}$ .

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

Step 8

Because there are no values to the left of $0$ in the domain of $\frac{\sqrt{x + h} - \sqrt{x}}{h}$ , the limit does not exist.

$Does not exist$

Step 9

Find the slope $m$ . In this case $m = D o e s (n o t) (e (1) i s t)$ .

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Step 9.1

Multiply $e$ by $1$ .

$m = D o e s (n o t) (e \cdot (1 i s t))$

Step 9.2

Remove parentheses.

$m = D o e s (n o t) (e (1) i s t)$

Step 10

The slope is $m = D o e s (n o t) (e (1) i s t)$ and the point is $(1, 1)$ .

$m = D o e s (n o t) (e (1) i s t)$

$m = (1, 1)$

Step 11

Multiply $e$ by $1$ .

$m = D o e s (n o t) (e \cdot (1 i s t))$

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.

$y = (D o e s (n o t) (e \cdot (1 i s t))) \cdot x + b$

Step 12.3

Substitute the value of $x$ into the equation.

$y = (D o e s (n o t) (e \cdot (1 i s t))) \cdot (1) + b$

Step 12.4

Substitute the value of $y$ into the equation.

$1 = (D o e s (n o t) (e \cdot (1 i s t))) \cdot (1) + b$

Step 12.5

Find the value of $b$ .

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Step 12.5.1

Rewrite the equation as $D o e s (n o t) (e \cdot (1 i s t)) \cdot 1 + b = 1$ .

$D o e s (n o t) (e \cdot (1 i s t)) \cdot 1 + b = 1$

Step 12.5.2

Simplify each term.

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Step 12.5.2.1

Multiply $o$ by $o$ by adding the exponents.

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Step 12.5.2.1.1

Move $o$ .

$D (o \cdot o) e s (n t) (e \cdot (1 i s t)) \cdot 1 + b = 1$

Step 12.5.2.1.2

Multiply $o$ by $o$ .

$D o^{2} e s (n t) (e \cdot (1 i s t)) \cdot 1 + b = 1$

Step 12.5.2.2

Simplify $D o^{2} e s (n t) (e \cdot (1 i s t)) \cdot 1$ .

$D o^{2} e s (n t) (e \cdot (i s t)) + b = 1$

Step 12.5.2.3

Multiply $s$ by $s$ by adding the exponents.

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Step 12.5.2.3.1

Move $s$ .

$D o^{2} e (s \cdot s) (n t) (e \cdot (i t)) + b = 1$

Step 12.5.2.3.2

Multiply $s$ by $s$ .

$D o^{2} e s^{2} (n t) (e \cdot (i t)) + b = 1$

Step 12.5.2.4

Multiply $t$ by $t$ by adding the exponents.

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Step 12.5.2.4.1

Move $t$ .

$D o^{2} e s^{2} (n (t \cdot t)) (e \cdot (i)) + b = 1$

Step 12.5.2.4.2

Multiply $t$ by $t$ .

$D o^{2} e s^{2} (n t^{2}) (e \cdot (i)) + b = 1$

Step 12.5.2.5

Multiply $D o^{2} e s^{2} n t^{2} (e i)$ .

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Step 12.5.2.5.1

Raise $e$ to the power of $1$ .

$D o^{2} s^{2} n t^{2} (e^{1} e i) + b = 1$

Step 12.5.2.5.2

Raise $e$ to the power of $1$ .

$D o^{2} s^{2} n t^{2} (e^{1} e^{1} i) + b = 1$

Step 12.5.2.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$D o^{2} s^{2} n t^{2} (e^{1 + 1} i) + b = 1$

Step 12.5.2.5.4

Add $1$ and $1$ .

$D o^{2} s^{2} n t^{2} (e^{2} i) + b = 1$

$D o^{2} s^{2} n t^{2} e^{2} i + b = 1$

Step 12.5.3

Subtract $D o^{2} s^{2} n t^{2} e^{2} i$ from both sides of the equation.

$b = 1 - D o^{2} s^{2} n t^{2} e^{2} i$

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.

$y = D o^{2} s^{2} n t^{2} e^{2} i x + 1 - D o^{2} s^{2} n t^{2} e^{2} i$

Step 14