Calculus Examples

$f (x) = - 3 \cos (x)$ at $x = \frac{3}{4} π$

Step 1

Find the corresponding $y$ -value to $x = \frac{3}{4} \cdot π$ .

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Step 1.1

Substitute $\frac{3}{4} \cdot π$ in for $x$ .

$y = - 3 \cos (\frac{3}{4} \cdot π)$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = - 3 \cos (\frac{3}{4} \cdot π)$

Step 1.2.2

Simplify $- 3 \cos (\frac{3}{4} \cdot π)$ .

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Step 1.2.2.1

Combine $\frac{3}{4}$ and $π$ .

$y = - 3 \cos (\frac{3 π}{4})$

Step 1.2.2.2

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

$y = - 3 (- \cos (\frac{π}{4}))$

Step 1.2.2.3

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

$y = - 3 (- \frac{\sqrt{2}}{2})$

Step 1.2.2.4

Multiply $- 3 (- \frac{\sqrt{2}}{2})$ .

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Step 1.2.2.4.1

Multiply $- 1$ by $- 3$ .

$y = 3 \frac{\sqrt{2}}{2}$

Step 1.2.2.4.2

Combine $3$ and $\frac{\sqrt{2}}{2}$ .

$y = \frac{3 \sqrt{2}}{2}$

Step 2

Find the first derivative and evaluate at $x = \frac{3}{4} π$ and $y = \frac{3 \sqrt{2}}{2}$ to find the slope of the tangent line.

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Step 2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 \cos (x)$ with respect to $x$ is $- 3 \frac{d}{d x} [\cos (x)]$ .

$- 3 \frac{d}{d x} [\cos (x)]$

Step 2.2

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

$- 3 (- \sin (x))$

Step 2.3

Multiply $- 1$ by $- 3$ .

$3 \sin (x)$

Step 2.4

Evaluate the derivative at $x = \frac{3}{4} \cdot π$ .

$3 \sin (\frac{3}{4} \cdot π)$

Step 2.5

Simplify.

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Step 2.5.1

Combine $\frac{3}{4}$ and $π$ .

$3 \sin (\frac{3 π}{4})$

Step 2.5.2

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

$3 \sin (\frac{π}{4})$

Step 2.5.3

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

$3 \frac{\sqrt{2}}{2}$

Step 2.5.4

Combine $3$ and $\frac{\sqrt{2}}{2}$ .

$\frac{3 \sqrt{2}}{2}$

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 $\frac{3 \sqrt{2}}{2}$ and a given point $(\frac{3}{4} \cdot π, \frac{3 \sqrt{2}}{2})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{3 \sqrt{2}}{2}) = \frac{3 \sqrt{2}}{2} \cdot (x - (\frac{3}{4} \cdot π))$

Step 3.2

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

$y - \frac{3 \sqrt{2}}{2} = \frac{3 \sqrt{2}}{2} \cdot (x - \frac{3 π}{4})$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{3 \sqrt{2}}{2} \cdot (x - \frac{3 π}{4})$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{3 \sqrt{2}}{2} = 0 + 0 + \frac{3 \sqrt{2}}{2} \cdot (x - \frac{3 π}{4})$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{3 \sqrt{2}}{2} = \frac{3 \sqrt{2}}{2} \cdot (x - \frac{3 π}{4})$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{3 \sqrt{2}}{2} = \frac{3 \sqrt{2}}{2} x + \frac{3 \sqrt{2}}{2} (- \frac{3 π}{4})$

Step 3.3.1.4

Combine $\frac{3 \sqrt{2}}{2}$ and $x$ .

$y - \frac{3 \sqrt{2}}{2} = \frac{3 \sqrt{2} x}{2} + \frac{3 \sqrt{2}}{2} (- \frac{3 π}{4})$

Step 3.3.1.5

Multiply $\frac{3 \sqrt{2}}{2} (- \frac{3 π}{4})$ .

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Step 3.3.1.5.1

Multiply $\frac{3 \sqrt{2}}{2}$ by $\frac{3 π}{4}$ .

$y - \frac{3 \sqrt{2}}{2} = \frac{3 \sqrt{2} x}{2} - \frac{3 \sqrt{2} (3 π)}{2 \cdot 4}$

Step 3.3.1.5.2

Multiply $3$ by $3$ .

$y - \frac{3 \sqrt{2}}{2} = \frac{3 \sqrt{2} x}{2} - \frac{9 \sqrt{2} π}{2 \cdot 4}$

Step 3.3.1.5.3

Multiply $2$ by $4$ .

$y - \frac{3 \sqrt{2}}{2} = \frac{3 \sqrt{2} x}{2} - \frac{9 \sqrt{2} π}{8}$

Step 3.3.2

Add $\frac{3 \sqrt{2}}{2}$ to both sides of the equation.

$y = \frac{3 \sqrt{2} x}{2} - \frac{9 \sqrt{2} π}{8} + \frac{3 \sqrt{2}}{2}$

Step 3.3.3

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

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Step 3.3.3.1

To write $\frac{3 \sqrt{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{3 \sqrt{2} x}{2} - \frac{9 \sqrt{2} π}{8} + \frac{3 \sqrt{2}}{2} \cdot \frac{4}{4}$

Step 3.3.3.2

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

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Step 3.3.3.2.1

Multiply $\frac{3 \sqrt{2}}{2}$ by $\frac{4}{4}$ .

$y = \frac{3 \sqrt{2} x}{2} - \frac{9 \sqrt{2} π}{8} + \frac{3 \sqrt{2} \cdot 4}{2 \cdot 4}$

Step 3.3.3.2.2

Multiply $2$ by $4$ .

$y = \frac{3 \sqrt{2} x}{2} - \frac{9 \sqrt{2} π}{8} + \frac{3 \sqrt{2} \cdot 4}{8}$

Step 3.3.3.3

Combine the numerators over the common denominator.

$y = \frac{3 \sqrt{2} x}{2} + \frac{- 9 \sqrt{2} π + 3 \sqrt{2} \cdot 4}{8}$

Step 3.3.3.4

Multiply $4$ by $3$ .

$y = \frac{3 \sqrt{2} x}{2} + \frac{- 9 \sqrt{2} π + 12 \sqrt{2}}{8}$

Step 3.3.3.5

Factor $- 1$ out of $- 9 \sqrt{2} π$ .

$y = \frac{3 \sqrt{2} x}{2} + \frac{- (9 \sqrt{2} π) + 12 \sqrt{2}}{8}$

Step 3.3.3.6

Factor $- 1$ out of $12 \sqrt{2}$ .

$y = \frac{3 \sqrt{2} x}{2} + \frac{- (9 \sqrt{2} π) - (- 12 \sqrt{2})}{8}$

Step 3.3.3.7

Factor $- 1$ out of $- (9 \sqrt{2} π) - (- 12 \sqrt{2})$ .

$y = \frac{3 \sqrt{2} x}{2} + \frac{- (9 \sqrt{2} π - 12 \sqrt{2})}{8}$

Step 3.3.3.8

Rewrite $- (9 \sqrt{2} π - 12 \sqrt{2})$ as $- 1 (9 \sqrt{2} π - 12 \sqrt{2})$ .

$y = \frac{3 \sqrt{2} x}{2} + \frac{- 1 (9 \sqrt{2} π - 12 \sqrt{2})}{8}$

Step 3.3.3.9

Move the negative in front of the fraction.

$y = \frac{3 \sqrt{2} x}{2} - \frac{9 \sqrt{2} π - 12 \sqrt{2}}{8}$

Step 3.3.3.10

Reorder terms.

$y = \frac{3 \sqrt{2}}{2} x - \frac{9 \sqrt{2} π - 12 \sqrt{2}}{8}$

Step 4