Calculus Examples

f (x) = 8 x^{2}

$f (x) = 8 x^{2}$ ,

16 x + y + 6 = 0

$16 x + y + 6 = 0$

Step 1

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 1.1

Subtract

16 x

$16 x$ from both sides of the equation.

y + 6 = - 16 x

$y + 6 = - 16 x$

Step 1.2

Subtract

6

$6$ from both sides of the equation.

y = - 16 x - 6

$y = - 16 x - 6$

y = - 16 x - 6

$y = - 16 x - 6$

Step 2

Use the slope-intercept form to find the slope.

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Step 2.1

The slope-intercept form is

y = m x + b

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 2.2

Using the slope-intercept form, the slope is

- 16

$- 16$ .

m = - 16

$m = - 16$

m = - 16

$m = - 16$

Step 3

Find the derivative.

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Step 3.1

Since

8

$8$ is constant with respect to

x

$x$ , the derivative of

8 x^{2}

$8 x^{2}$ with respect to

x

$x$ is

8 \frac{d}{d x} [x^{2}]

$8 \frac{d}{d x} [x^{2}]$ .

8 \frac{d}{d x} [x^{2}]

$8 \frac{d}{d x} [x^{2}]$

Step 3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

8 (2 x)

$8 (2 x)$

Step 3.3

Multiply

2

$2$ by

8

$8$ .

16 x

$16 x$

16 x

$16 x$

Step 4

The first derivative of a function represents the slope at every point of that function. In this case, the derivative of

f (x) = 8 x^{2}

$f (x) = 8 x^{2}$ is

16 x

$16 x$ and the slope of the given line

y = - 16 x - 6

$y = - 16 x - 6$ is

m = - 16

$m = - 16$ . To find the point on

f (x) = 8 x^{2}

$f (x) = 8 x^{2}$ where the slope of the tangent line is the same as the slope of the given line

y = - 16 x - 6

$y = - 16 x - 6$ , substitute the value of the slope of the given line

- 16

$- 16$ for the value of

16 x

$16 x$ .

- 16 = 16 x

$- 16 = 16 x$

Step 5

Solve

- 16 = 16 x

$- 16 = 16 x$ for

x

$x$ to find the x-coordinate of the point where the tangent line is parallel to the given line

y = - 16 x - 6

$y = - 16 x - 6$ . In this case, the x-coordinate is

x = - 1

$x = - 1$ .

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Step 5.1

Rewrite the equation as

16 x = - 16

$16 x = - 16$ .

16 x = - 16

$16 x = - 16$

Step 5.2

Divide each term in

16 x = - 16

$16 x = - 16$ by

16

$16$ and simplify.

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Step 5.2.1

Divide each term in

16 x = - 16

$16 x = - 16$ by

16

$16$ .

\frac{16 x}{16} = \frac{- 16}{16}

$\frac{16 x}{16} = \frac{- 16}{16}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

16

$16$ .

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Step 5.2.2.1.1

Cancel the common factor.

\frac{16 x}{16} = \frac{- 16}{16}

$\frac{16 x}{16} = \frac{- 16}{16}$

Step 5.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- 16}{16}

$x = \frac{- 16}{16}$

x = \frac{- 16}{16}

$x = \frac{- 16}{16}$

x = \frac{- 16}{16}

$x = \frac{- 16}{16}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide

- 16

$- 16$ by

16

$16$ .

x = - 1

$x = - 1$

x = - 1

$x = - 1$

x = - 1

$x = - 1$

x = - 1

$x = - 1$

Step 6

Substitute

- 1

$- 1$ in

f (x) = 8 x^{2}

$f (x) = 8 x^{2}$ to get the y-coordinate of the point where the tangent line is parallel to the given line

y = - 16 x - 6

$y = - 16 x - 6$ . In this case, the y-coordinate is

8

$8$ .

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Step 6.1

Replace the variable

x

$x$ with

- 1

$- 1$ in the expression.

f (- 1) = 8 {(- 1)}^{2}

$f (- 1) = 8 {(- 1)}^{2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Raise

- 1

$- 1$ to the power of

2

$2$ .

f (- 1) = 8 \cdot 1

$f (- 1) = 8 \cdot 1$

Step 6.2.2

Multiply

8

$8$ by

1

$1$ .

f (- 1) = 8

$f (- 1) = 8$

Step 6.2.3

The final answer is

8

$8$ .

8

$8$

8

$8$

8

$8$

Step 7

The point on

f (x) = 8 x^{2}

$f (x) = 8 x^{2}$ where the slope of the tangent line is the same as the slope of the given line

y = - 16 x - 6

$y = - 16 x - 6$ has x-coordinate of

- 1

$- 1$ and y-coordinate of

8

$8$ . The slope of the tangent line is the same as the slope of

y = - 16 x - 6

$y = - 16 x - 6$ , which is

m = - 16

$m = - 16$ .

(- 1, 8), m = - 16

$(- 1, 8), m = - 16$

Step 8

The tangent line on

(- 1, 8)

$(- 1, 8)$ where the slope is

m = - 16

$m = - 16$ .

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Step 8.1

Find the value of

b

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

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Step 8.1.1

Use the formula for the equation of a line to find

b

$b$ .

y = m x + b

$y = m x + b$

Step 8.1.2

Substitute the value of

m

$m$ into the equation.

y = (- 16) \cdot x + b

$y = (- 16) \cdot x + b$

Step 8.1.3

Substitute the value of

x

$x$ into the equation.

y = (- 16) \cdot (- 1) + b

$y = (- 16) \cdot (- 1) + b$

Step 8.1.4

Substitute the value of

y

$y$ into the equation.

8 = (- 16) \cdot (- 1) + b

$8 = (- 16) \cdot (- 1) + b$

Step 8.1.5

Find the value of

b

$b$ .

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Step 8.1.5.1

Rewrite the equation as

(- 16) \cdot (- 1) + b = 8

$(- 16) \cdot (- 1) + b = 8$ .

(- 16) \cdot (- 1) + b = 8

$(- 16) \cdot (- 1) + b = 8$

Step 8.1.5.2

Multiply

- 16

$- 16$ by

- 1

$- 1$ .

16 + b = 8

$16 + b = 8$

Step 8.1.5.3

Move all terms not containing

b

$b$ to the right side of the equation.

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Step 8.1.5.3.1

Subtract

16

$16$ from both sides of the equation.

b = 8 - 16

$b = 8 - 16$

Step 8.1.5.3.2

Subtract

16

$16$ from

8

$8$ .

b = - 8

$b = - 8$

b = - 8

$b = - 8$

b = - 8

$b = - 8$

b = - 8

$b = - 8$

Step 8.2

Now that the values of

m

$m$ (slope) and

b

$b$ (y-intercept) are known, substitute them into

y = m x + b

$y = m x + b$ to find the equation of the line.

y = - 16 x - 8

$y = - 16 x - 8$

y = - 16 x - 8

$y = - 16 x - 8$

Step 9