Calculus Examples

$f (x) = - 2 - \cos (x)$ at $x = - \frac{π}{4}$

Step 1

Find the corresponding $y$ -value to $x = - \frac{π}{4}$ .

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Step 1.1

Substitute $- \frac{π}{4}$ in for $x$ .

$y = - 2 - \cos (- \frac{π}{4})$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = - 2 - \cos (- \frac{π}{4})$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$y = - 2 - \cos (\frac{7 π}{4})$

Step 1.2.2.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$y = - 2 - \cos (\frac{π}{4})$

Step 1.2.2.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$y = - 2 - \frac{\sqrt{2}}{2}$

Step 2

Find the first derivative and evaluate at $x = - \frac{π}{4}$ and $y = - 2 - \frac{\sqrt{2}}{2}$ to find the slope of the tangent line.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $- 2 - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [- 2] + \frac{d}{d x} [- \cos (x)]$ .

$\frac{d}{d x} [- 2] + \frac{d}{d x} [- \cos (x)]$

Step 2.1.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \cos (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

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Step 2.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

$0 - \frac{d}{d x} [\cos (x)]$

Step 2.2.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$0 - - \sin (x)$

Step 2.2.3

Multiply $- 1$ by $- 1$ .

$0 + 1 \sin (x)$

Step 2.2.4

Multiply $\sin (x)$ by $1$ .

$0 + \sin (x)$

Step 2.3

Add $0$ and $\sin (x)$ .

$\sin (x)$

Step 2.4

Evaluate the derivative at $x = - \frac{π}{4}$ .

$\sin (- \frac{π}{4})$

Step 2.5

Simplify.

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Step 2.5.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sin (\frac{7 π}{4})$

Step 2.5.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- \sin (\frac{π}{4})$

Step 2.5.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2}$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $- \frac{\sqrt{2}}{2}$ and a given point $(- \frac{π}{4}, - 2 - \frac{\sqrt{2}}{2})$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2 - \frac{\sqrt{2}}{2}) = - \frac{\sqrt{2}}{2} \cdot (x - (- \frac{π}{4}))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y + 2 + \frac{\sqrt{2}}{2} = - \frac{\sqrt{2}}{2} \cdot (x + \frac{π}{4})$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- \frac{\sqrt{2}}{2} \cdot (x + \frac{π}{4})$ .

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Step 3.3.1.1

Rewrite.

$y + 2 + \frac{\sqrt{2}}{2} = 0 + 0 - \frac{\sqrt{2}}{2} \cdot (x + \frac{π}{4})$

Step 3.3.1.2

Simplify by adding zeros.

$y + 2 + \frac{\sqrt{2}}{2} = - \frac{\sqrt{2}}{2} \cdot (x + \frac{π}{4})$

Step 3.3.1.3

Apply the distributive property.

$y + 2 + \frac{\sqrt{2}}{2} = - \frac{\sqrt{2}}{2} x - \frac{\sqrt{2}}{2} \cdot \frac{π}{4}$

Step 3.3.1.4

Combine $x$ and $\frac{\sqrt{2}}{2}$ .

$y + 2 + \frac{\sqrt{2}}{2} = - \frac{x \sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{π}{4}$

Step 3.3.1.5

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{π}{4}$ .

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Step 3.3.1.5.1

Multiply $\frac{π}{4}$ by $\frac{\sqrt{2}}{2}$ .

$y + 2 + \frac{\sqrt{2}}{2} = - \frac{x \sqrt{2}}{2} - \frac{π \sqrt{2}}{4 \cdot 2}$

Step 3.3.1.5.2

Multiply $4$ by $2$ .

$y + 2 + \frac{\sqrt{2}}{2} = - \frac{x \sqrt{2}}{2} - \frac{π \sqrt{2}}{8}$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $2$ from both sides of the equation.

$y + \frac{\sqrt{2}}{2} = - \frac{x \sqrt{2}}{2} - \frac{π \sqrt{2}}{8} - 2$

Step 3.3.2.2

Subtract $\frac{\sqrt{2}}{2}$ from both sides of the equation.

$y = - \frac{x \sqrt{2}}{2} - \frac{π \sqrt{2}}{8} - 2 - \frac{\sqrt{2}}{2}$

Step 3.3.3

Write in $y = m x + b$ form.

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Step 3.3.3.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = - \frac{x \sqrt{2}}{2} - \frac{π \sqrt{2}}{8} - 2 \cdot \frac{8}{8} - \frac{\sqrt{2}}{2}$

Step 3.3.3.2

Combine $- 2$ and $\frac{8}{8}$ .

$y = - \frac{x \sqrt{2}}{2} - \frac{π \sqrt{2}}{8} + \frac{- 2 \cdot 8}{8} - \frac{\sqrt{2}}{2}$

Step 3.3.3.3

Combine the numerators over the common denominator.

$y = - \frac{x \sqrt{2}}{2} + \frac{- π \sqrt{2} - 2 \cdot 8}{8} - \frac{\sqrt{2}}{2}$

Step 3.3.3.4

Multiply $- 2$ by $8$ .

$y = - \frac{x \sqrt{2}}{2} + \frac{- π \sqrt{2} - 16}{8} - \frac{\sqrt{2}}{2}$

Step 3.3.3.5

To write $- \frac{\sqrt{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - \frac{x \sqrt{2}}{2} + \frac{- π \sqrt{2} - 16}{8} - \frac{\sqrt{2}}{2} \cdot \frac{4}{4}$

Step 3.3.3.6

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.3.6.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{4}{4}$ .

$y = - \frac{x \sqrt{2}}{2} + \frac{- π \sqrt{2} - 16}{8} - \frac{\sqrt{2} \cdot 4}{2 \cdot 4}$

Step 3.3.3.6.2

Multiply $2$ by $4$ .

$y = - \frac{x \sqrt{2}}{2} + \frac{- π \sqrt{2} - 16}{8} - \frac{\sqrt{2} \cdot 4}{8}$

Step 3.3.3.7

Combine the numerators over the common denominator.

$y = - \frac{x \sqrt{2}}{2} + \frac{- π \sqrt{2} - 16 - \sqrt{2} \cdot 4}{8}$

Step 3.3.3.8

Multiply $4$ by $- 1$ .

$y = - \frac{x \sqrt{2}}{2} + \frac{- π \sqrt{2} - 16 - 4 \sqrt{2}}{8}$

Step 3.3.3.9

Factor $- 1$ out of $- π \sqrt{2}$ .

$y = - \frac{x \sqrt{2}}{2} + \frac{- (π \sqrt{2}) - 16 - 4 \sqrt{2}}{8}$

Step 3.3.3.10

Rewrite $- 16$ as $- 1 (16)$ .

$y = - \frac{x \sqrt{2}}{2} + \frac{- (π \sqrt{2}) - 1 (16) - 4 \sqrt{2}}{8}$

Step 3.3.3.11

Factor $- 1$ out of $- (π \sqrt{2}) - 1 (16)$ .

$y = - \frac{x \sqrt{2}}{2} + \frac{- (π \sqrt{2} + 16) - 4 \sqrt{2}}{8}$

Step 3.3.3.12

Factor $- 1$ out of $- 4 \sqrt{2}$ .

$y = - \frac{x \sqrt{2}}{2} + \frac{- (π \sqrt{2} + 16) - (4 \sqrt{2})}{8}$

Step 3.3.3.13

Factor $- 1$ out of $- (π \sqrt{2} + 16) - (4 \sqrt{2})$ .

$y = - \frac{x \sqrt{2}}{2} + \frac{- (π \sqrt{2} + 16 + 4 \sqrt{2})}{8}$

Step 3.3.3.14

Rewrite $- (π \sqrt{2} + 16 + 4 \sqrt{2})$ as $- 1 (π \sqrt{2} + 16 + 4 \sqrt{2})$ .

$y = - \frac{x \sqrt{2}}{2} + \frac{- 1 (π \sqrt{2} + 16 + 4 \sqrt{2})}{8}$

Step 3.3.3.15

Move the negative in front of the fraction.

$y = - \frac{x \sqrt{2}}{2} - \frac{π \sqrt{2} + 16 + 4 \sqrt{2}}{8}$

Step 3.3.3.16

Reorder terms.

$y = - (\frac{\sqrt{2}}{2} x) - \frac{π \sqrt{2} + 16 + 4 \sqrt{2}}{8}$

Step 3.3.3.17

Remove parentheses.

$y = - \frac{\sqrt{2}}{2} x - \frac{π \sqrt{2} + 16 + 4 \sqrt{2}}{8}$

Step 4