Calculus Examples

$f (x) = \sqrt{x}$ , $(3, \sqrt{3})$

Step 1

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

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Step 1.1

Evaluate $f (x) = \sqrt{x}$ at $x = 3$ .

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Step 1.1.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \sqrt{3}$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Remove parentheses.

$f (3) = \sqrt{3}$

Step 1.1.2.2

The final answer is $\sqrt{3}$ .

$\sqrt{3}$

Step 1.2

Since $\sqrt{3} = \sqrt{3}$ , 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 $f (x) = \sqrt{x}$

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.

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Remove parentheses.

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

Step 4.1.2.2

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

$\sqrt{x + h}$

Step 4.2

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 5

Plug in the components.

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

Step 6

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

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

Step 7

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 8

Find the slope $m$ . In this case $m = D o e s (n o t) (e (3) i s t)$ .

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Step 8.1

Multiply $e$ by $3$ .

$m = D o e s (n o t) (e \cdot (3 i s t))$

Step 8.2

Remove parentheses.

$m = D o e s (n o t) (e (3) i s t)$

Step 9

The slope is $m = D o e s (n o t) (e (3) i s t)$ and the point is $(3, \sqrt{3})$ .

$m = D o e s (n o t) (e (3) i s t)$

$m = (3, \sqrt{3})$

Step 10

Multiply $e$ by $3$ .

$m = D o e s (n o t) (e \cdot (3 i s t))$

Step 11

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

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Step 11.1

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

$y = m x + b$

Step 11.2

Substitute the value of $m$ into the equation.

$y = (D o e s (n o t) (e \cdot (3 i s t))) \cdot x + b$

Step 11.3

Substitute the value of $x$ into the equation.

$y = (D o e s (n o t) (e \cdot (3 i s t))) \cdot (3) + b$

Step 11.4

Substitute the value of $y$ into the equation.

$\sqrt{3} = (D o e s (n o t) (e \cdot (3 i s t))) \cdot (3) + b$

Step 11.5

Find the value of $b$ .

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Step 11.5.1

Rewrite the equation as $D o e s (n o t) (e \cdot (3 i s t)) \cdot 3 + b = \sqrt{3}$ .

$D o e s (n o t) (e \cdot (3 i s t)) \cdot 3 + b = \sqrt{3}$

Step 11.5.2

Simplify each term.

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Step 11.5.2.1

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

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Step 11.5.2.1.1

Move $o$ .

$D (o \cdot o) e s (n t) (e \cdot (3 i s t)) \cdot 3 + b = \sqrt{3}$

Step 11.5.2.1.2

Multiply $o$ by $o$ .

$D o^{2} e s (n t) (e \cdot (3 i s t)) \cdot 3 + b = \sqrt{3}$

Step 11.5.2.2

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

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Step 11.5.2.2.1

Move $s$ .

$D o^{2} e (s \cdot s) (n t) (e \cdot (3 i t)) \cdot 3 + b = \sqrt{3}$

Step 11.5.2.2.2

Multiply $s$ by $s$ .

$D o^{2} e s^{2} (n t) (e \cdot (3 i t)) \cdot 3 + b = \sqrt{3}$

Step 11.5.2.3

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

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Step 11.5.2.3.1

Move $t$ .

$D o^{2} e s^{2} (n (t \cdot t)) (e \cdot (3 i)) \cdot 3 + b = \sqrt{3}$

Step 11.5.2.3.2

Multiply $t$ by $t$ .

$D o^{2} e s^{2} (n t^{2}) (e \cdot (3 i)) \cdot 3 + b = \sqrt{3}$

Step 11.5.2.4

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

$D o^{2} e s^{2} n t^{2} (3 \cdot e i) \cdot 3 + b = \sqrt{3}$

Step 11.5.2.5

Multiply $D o^{2} e s^{2} n t^{2} (3 e i)$ .

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Step 11.5.2.5.1

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

$D o^{2} s^{2} n t^{2} (3 (e^{1} e) i) \cdot 3 + b = \sqrt{3}$

Step 11.5.2.5.2

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

$D o^{2} s^{2} n t^{2} (3 (e^{1} e^{1}) i) \cdot 3 + b = \sqrt{3}$

Step 11.5.2.5.3

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

$D o^{2} s^{2} n t^{2} (3 e^{1 + 1} i) \cdot 3 + b = \sqrt{3}$

Step 11.5.2.5.4

Add $1$ and $1$ .

$D o^{2} s^{2} n t^{2} (3 e^{2} i) \cdot 3 + b = \sqrt{3}$

Step 11.5.2.6

Move $3$ to the left of $D o^{2} s^{2} n t^{2}$ .

$3 \cdot (D o^{2} s^{2} n t^{2}) e^{2} i \cdot 3 + b = \sqrt{3}$

Step 11.5.2.7

Multiply $3$ by $3$ .

$9 D o^{2} s^{2} n t^{2} e^{2} i + b = \sqrt{3}$

Step 11.5.3

Subtract $9 D o^{2} s^{2} n t^{2} e^{2} i$ from both sides of the equation.

$b = \sqrt{3} - 9 D o^{2} s^{2} n t^{2} e^{2} i$

Step 12

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.

$y = 3 D o^{2} s^{2} n t^{2} e^{2} i x + \sqrt{3} - 9 D o^{2} s^{2} n t^{2} e^{2} i$

Step 13