Calculus Examples

7 x^{2} + 3 x

$7 x^{2} + 3 x$ ,

(1, 10)

$(1, 10)$

Step 1

Write

7 x^{2} + 3 x

$7 x^{2} + 3 x$ as a function.

f (x) = 7 x^{2} + 3 x

$f (x) = 7 x^{2} + 3 x$

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) = 7 x^{2} + 3 x

$f (x) = 7 x^{2} + 3 x$ at

x = 1

$x = 1$ .

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Step 2.1.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

f (1) = 7 {(1)}^{2} + 3 (1)

$f (1) = 7 {(1)}^{2} + 3 (1)$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

One to any power is one.

f (1) = 7 \cdot 1 + 3 (1)

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

Step 2.1.2.1.2

Multiply

7

$7$ by

1

$1$ .

f (1) = 7 + 3 (1)

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

Step 2.1.2.1.3

Multiply

3

$3$ by

1

$1$ .

f (1) = 7 + 3

$f (1) = 7 + 3$

f (1) = 7 + 3

$f (1) = 7 + 3$

Step 2.1.2.2

Add

7

$7$ and

3

$3$ .

f (1) = 10

$f (1) = 10$

Step 2.1.2.3

The final answer is

10

$10$ .

10

$10$

10

$10$

10

$10$

Step 2.2

Since

10 = 10

$10 = 10$ , 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

$m$

=

$=$ The derivative of

f (x) = 7 x^{2} + 3 x

$f (x) = 7 x^{2} + 3 x$

Step 4

Consider the limit definition of the derivative.

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

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

Step 5

Find the components of the definition.

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Step 5.1

Evaluate the function at

x = x + h

$x = x + h$ .

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Step 5.1.1

Replace the variable

x

$x$ with

x + h

$x + h$ in the expression.

f (x + h) = 7 {(x + h)}^{2} + 3 (x + h)

$f (x + h) = 7 {(x + h)}^{2} + 3 (x + h)$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

Rewrite

{(x + h)}^{2}

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

(x + h) (x + h)

$(x + h) (x + h)$ .

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

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

Step 5.1.2.1.2

Expand

(x + h) (x + h)

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

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Step 5.1.2.1.2.1

Apply the distributive property.

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

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

Step 5.1.2.1.2.2

Apply the distributive property.

f (x + h) = 7 (x \cdot x + x h + h (x + h)) + 3 (x + h)

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

Step 5.1.2.1.2.3

Apply the distributive property.

f (x + h) = 7 (x \cdot x + x h + h x + h \cdot h) + 3 (x + h)

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

f (x + h) = 7 (x \cdot x + x h + h x + h \cdot h) + 3 (x + h)

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

Step 5.1.2.1.3

Simplify and combine like terms.

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Step 5.1.2.1.3.1

Simplify each term.

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Step 5.1.2.1.3.1.1

Multiply

x

$x$ by

x

$x$ .

f (x + h) = 7 (x^{2} + x h + h x + h \cdot h) + 3 (x + h)

$f (x + h) = 7 (x^{2} + x h + h x + h \cdot h) + 3 (x + h)$

Step 5.1.2.1.3.1.2

Multiply

h

$h$ by

h

$h$ .

f (x + h) = 7 (x^{2} + x h + h x + h^{2}) + 3 (x + h)

$f (x + h) = 7 (x^{2} + x h + h x + h^{2}) + 3 (x + h)$

f (x + h) = 7 (x^{2} + x h + h x + h^{2}) + 3 (x + h)

$f (x + h) = 7 (x^{2} + x h + h x + h^{2}) + 3 (x + h)$

Step 5.1.2.1.3.2

Add

x h

$x h$ and

h x

$h x$ .

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Step 5.1.2.1.3.2.1

Reorder

x

$x$ and

h

$h$ .

f (x + h) = 7 (x^{2} + h x + h x + h^{2}) + 3 (x + h)

$f (x + h) = 7 (x^{2} + h x + h x + h^{2}) + 3 (x + h)$

Step 5.1.2.1.3.2.2

Add

h x

$h x$ and

h x

$h x$ .

f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 3 (x + h)

$f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 3 (x + h)$

f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 3 (x + h)

$f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 3 (x + h)$

f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 3 (x + h)

$f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 3 (x + h)$

Step 5.1.2.1.4

Apply the distributive property.

f (x + h) = 7 x^{2} + 7 (2 h x) + 7 h^{2} + 3 (x + h)

$f (x + h) = 7 x^{2} + 7 (2 h x) + 7 h^{2} + 3 (x + h)$

Step 5.1.2.1.5

Multiply

2

$2$ by

7

$7$ .

f (x + h) = 7 x^{2} + 14 (h x) + 7 h^{2} + 3 (x + h)

$f (x + h) = 7 x^{2} + 14 (h x) + 7 h^{2} + 3 (x + h)$

Step 5.1.2.1.6

Apply the distributive property.

f (x + h) = 7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h

$f (x + h) = 7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h$

f (x + h) = 7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h

$f (x + h) = 7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h$

Step 5.1.2.2

The final answer is

7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h

$7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h$ .

7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h

$7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h$

7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h

$7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h$

7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h

$7 x^{2} + 14 h x + 7 h^{2} + 3 x + 3 h$

Step 5.2

Reorder.

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Step 5.2.1

Move

3 x

$3 x$ .

7 x^{2} + 14 h x + 7 h^{2} + 3 h + 3 x

$7 x^{2} + 14 h x + 7 h^{2} + 3 h + 3 x$

Step 5.2.2

Move

7 x^{2}

$7 x^{2}$ .

14 h x + 7 h^{2} + 7 x^{2} + 3 h + 3 x

$14 h x + 7 h^{2} + 7 x^{2} + 3 h + 3 x$

Step 5.2.3

Reorder

14 h x

$14 h x$ and

7 h^{2}

$7 h^{2}$ .

7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x

$7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x$

7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x

$7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x$

Step 5.3

Find the components of the definition.

f (x + h) = 7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x

$f (x + h) = 7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x$

f (x) = 7 x^{2} + 3 x

$f (x) = 7 x^{2} + 3 x$

f (x + h) = 7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x

$f (x + h) = 7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x$

f (x) = 7 x^{2} + 3 x

$f (x) = 7 x^{2} + 3 x$

Step 6

Plug in the components.

f' (x) = lim h \to 0 \frac{7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x - (7 x^{2} + 3 x)}{h}

$f' (x) = lim_{h \to 0} \frac{7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x - (7 x^{2} + 3 x)}{h}$

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{7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x - (7 x^{2}) - (3 x)}{h}

$f' (x) = lim_{h \to 0} \frac{7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x - (7 x^{2}) - (3 x)}{h}$

Step 7.1.2

Multiply

7

$7$ by

- 1

$- 1$ .

f' (x) = lim h \to 0 \frac{7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x - 7 x^{2} - (3 x)}{h}

$f' (x) = lim_{h \to 0} \frac{7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x - 7 x^{2} - (3 x)}{h}$

Step 7.1.3

Multiply

3

$3$ by

- 1

$- 1$ .

f' (x) = lim h \to 0 \frac{7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x - 7 x^{2} - 3 x}{h}

$f' (x) = lim_{h \to 0} \frac{7 h^{2} + 14 h x + 7 x^{2} + 3 h + 3 x - 7 x^{2} - 3 x}{h}$

Step 7.1.4

Subtract

7 x^{2}

$7 x^{2}$ from

7 x^{2}

$7 x^{2}$ .

f' (x) = lim h \to 0 \frac{7 h^{2} + 14 h x + 3 h + 3 x + 0 - 3 x}{h}

$f' (x) = lim_{h \to 0} \frac{7 h^{2} + 14 h x + 3 h + 3 x + 0 - 3 x}{h}$

Step 7.1.5

Add

7 h^{2}

$7 h^{2}$ and

0

$0$ .

f' (x) = lim h \to 0 \frac{7 h^{2} + 14 h x + 3 h + 3 x - 3 x}{h}

$f' (x) = lim_{h \to 0} \frac{7 h^{2} + 14 h x + 3 h + 3 x - 3 x}{h}$

Step 7.1.6

Subtract

3 x

$3 x$ from

3 x

$3 x$ .

f' (x) = lim h \to 0 \frac{7 h^{2} + 14 h x + 3 h + 0}{h}

$f' (x) = lim_{h \to 0} \frac{7 h^{2} + 14 h x + 3 h + 0}{h}$

Step 7.1.7

Add

7 h^{2} + 14 h x + 3 h

$7 h^{2} + 14 h x + 3 h$ and

0

$0$ .

f' (x) = lim h \to 0 \frac{7 h^{2} + 14 h x + 3 h}{h}

$f' (x) = lim_{h \to 0} \frac{7 h^{2} + 14 h x + 3 h}{h}$

Step 7.1.8

Factor

h

$h$ out of

7 h^{2} + 14 h x + 3 h

$7 h^{2} + 14 h x + 3 h$ .

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Step 7.1.8.1

Factor

h

$h$ out of

7 h^{2}

$7 h^{2}$ .

f' (x) = lim h \to 0 \frac{h (7 h) + 14 h x + 3 h}{h}

$f' (x) = lim_{h \to 0} \frac{h (7 h) + 14 h x + 3 h}{h}$

Step 7.1.8.2

Factor

h

$h$ out of

14 h x

$14 h x$ .

f' (x) = lim h \to 0 \frac{h (7 h) + h (14 x) + 3 h}{h}

$f' (x) = lim_{h \to 0} \frac{h (7 h) + h (14 x) + 3 h}{h}$

Step 7.1.8.3

Factor

h

$h$ out of

3 h

$3 h$ .

f' (x) = lim h \to 0 \frac{h (7 h) + h (14 x) + h \cdot 3}{h}

$f' (x) = lim_{h \to 0} \frac{h (7 h) + h (14 x) + h \cdot 3}{h}$

Step 7.1.8.4

Factor

h

$h$ out of

h (7 h) + h (14 x)

$h (7 h) + h (14 x)$ .

f' (x) = lim h \to 0 \frac{h (7 h + 14 x) + h \cdot 3}{h}

$f' (x) = lim_{h \to 0} \frac{h (7 h + 14 x) + h \cdot 3}{h}$

Step 7.1.8.5

Factor

h

$h$ out of

h (7 h + 14 x) + h \cdot 3

$h (7 h + 14 x) + h \cdot 3$ .

f' (x) = lim h \to 0 \frac{h (7 h + 14 x + 3)}{h}

$f' (x) = lim_{h \to 0} \frac{h (7 h + 14 x + 3)}{h}$

f' (x) = lim h \to 0 \frac{h (7 h + 14 x + 3)}{h}

$f' (x) = lim_{h \to 0} \frac{h (7 h + 14 x + 3)}{h}$

f' (x) = lim h \to 0 \frac{h (7 h + 14 x + 3)}{h}

$f' (x) = lim_{h \to 0} \frac{h (7 h + 14 x + 3)}{h}$

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

$h$ .

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Step 7.2.1.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h (7 h + 14 x + 3)}{h}$

Step 7.2.1.2

Divide

$7 h + 14 x + 3$ by

$1$ .

$f' (x) = lim_{h \to 0} 7 h + 14 x + 3$

Step 7.2.2

Reorder

$7 h$ and

$14 x$ .

$f' (x) = lim_{h \to 0} 14 x + 7 h + 3$

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

$0$ .

$lim_{h \to 0} 14 x + lim_{h \to 0} 7 h + lim_{h \to 0} 3$

Step 8.2

Evaluate the limit of

$14 x$ which is constant as

$h$ approaches

$0$ .

$14 x + lim_{h \to 0} 7 h + lim_{h \to 0} 3$

Step 8.3

Move the term

$7$ outside of the limit because it is constant with respect to

$h$ .

$14 x + 7 lim_{h \to 0} h + lim_{h \to 0} 3$

Step 8.4

Evaluate the limit of

$3$ which is constant as

$h$ approaches

$0$ .

$14 x + 7 lim_{h \to 0} h + 3$

Step 9

Evaluate the limit of

$h$ by plugging in

$0$ for

$h$ .

$14 x + 7 \cdot 0 + 3$

Step 10

Simplify the answer.

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Step 10.1

Multiply

$7$ by

$0$ .

$14 x + 0 + 3$

Step 10.2

Add

$14 x$ and

$0$ .

$14 x + 3$

Step 11

Simplify

$14 (1) + 3$ .

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Step 11.1

Multiply

$14$ by

$1$ .

$m = 14 + 3$

Step 11.2

Add

$14$ and

$3$ .

$m = 17$

Step 12

The slope is

$m = 17$ and the point is

$(1, 10)$ .

$m = 17, (1, 10)$

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.

$y = (17) \cdot x + b$

Step 13.3

Substitute the value of

$x$ into the equation.

$y = (17) \cdot (1) + b$

Step 13.4

Substitute the value of

$y$ into the equation.

$10 = (17) \cdot (1) + b$

Step 13.5

Find the value of

$b$ .

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Step 13.5.1

Rewrite the equation as

$(17) \cdot (1) + b = 10$ .

$(17) \cdot (1) + b = 10$

Step 13.5.2

Multiply

$17$ by

$1$ .

$17 + b = 10$

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

$17$ from both sides of the equation.

$b = 10 - 17$

Step 13.5.3.2

Subtract

$17$ from

$10$ .

$b = - 7$

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.

$y = 17 x - 7$

Step 15

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