Calculus Examples

$f (x) = 7 x + 3$ ; $(1, 10)$

Step 1

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

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Step 1.1

Evaluate $f (x) = 7 x + 3$ at $x = 1$ .

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Step 1.1.1

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

$f (1) = 7 (1) + 3$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Multiply $7$ by $1$ .

$f (1) = 7 + 3$

Step 1.1.2.2

Add $7$ and $3$ .

$f (1) = 10$

Step 1.1.2.3

The final answer is $10$ .

$10$

Step 1.2

Since $10 = 10$ , 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 $f (x) = 7 x + 3$

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.

$f (x + h) = 7 (x + h) + 3$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Apply the distributive property.

$f (x + h) = 7 x + 7 h + 3$

Step 4.1.2.2

The final answer is $7 x + 7 h + 3$ .

$7 x + 7 h + 3$

Step 4.2

Reorder $7 x$ and $7 h$ .

$7 h + 7 x + 3$

Step 4.3

Find the components of the definition.

$f (x + h) = 7 h + 7 x + 3$

$f (x) = 7 x + 3$

$f (x + h) = 7 h + 7 x + 3$

$f (x) = 7 x + 3$

Step 5

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{7 h + 7 x + 3 - (7 x + 3)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{7 h + 7 x + 3 - (7 x) - 1 \cdot 3}{h}$

Step 6.1.2

Multiply $7$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{7 h + 7 x + 3 - 7 x - 1 \cdot 3}{h}$

Step 6.1.3

Multiply $- 1$ by $3$ .

$f' (x) = lim_{h \to 0} \frac{7 h + 7 x + 3 - 7 x - 3}{h}$

Step 6.1.4

Subtract $7 x$ from $7 x$ .

$f' (x) = lim_{h \to 0} \frac{7 h + 0 + 3 - 3}{h}$

Step 6.1.5

Add $7 h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{7 h + 3 - 3}{h}$

Step 6.1.6

Subtract $3$ from $3$ .

$f' (x) = lim_{h \to 0} \frac{7 h + 0}{h}$

Step 6.1.7

Add $7 h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{7 h}{h}$

Step 6.2

Cancel the common factor of $h$ .

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Step 6.2.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{7 h}{h}$

Step 6.2.2

Divide $7$ by $1$ .

$f' (x) = lim_{h \to 0} 7$

Step 7

Evaluate the limit of $7$ which is constant as $h$ approaches $0$ .

$7$

Step 8

The slope is and the point is $(1, 10)$ .

$m = 7, (1, 10)$

Step 9

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

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Step 9.1

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

$y = m x + b$

Step 9.2

Substitute the value of $m$ into the equation.

$y = (7) \cdot x + b$

Step 9.3

Substitute the value of $x$ into the equation.

$y = (7) \cdot (1) + b$

Step 9.4

Substitute the value of $y$ into the equation.

$10 = (7) \cdot (1) + b$

Step 9.5

Find the value of $b$ .

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Step 9.5.1

Rewrite the equation as $(7) \cdot (1) + b = 10$ .

$(7) \cdot (1) + b = 10$

Step 9.5.2

Multiply $7$ by $1$ .

$7 + b = 10$

Step 9.5.3

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

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Step 9.5.3.1

Subtract $7$ from both sides of the equation.

$b = 10 - 7$

Step 9.5.3.2

Subtract $7$ from $10$ .

$b = 3$

Step 10

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.

$y = 7 x + 3$

Step 11