Calculus Examples

$f (x) = - 2 \sin (x)$ 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 $x$ .

$y = - 2 \sin (\frac{3}{4} \cdot π)$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = - 2 \sin (\frac{3}{4} \cdot π)$

Step 1.2.2

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

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Step 1.2.2.1

Combine $\frac{3}{4}$ and $π$ .

$y = - 2 \sin (\frac{3 π}{4})$

Step 1.2.2.2

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

$y = - 2 \sin (\frac{π}{4})$

Step 1.2.2.3

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

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

Step 1.2.2.4

Cancel the common factor of $2$ .

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Step 1.2.2.4.1

Factor $2$ out of $- 2$ .

$y = 2 (- 1) \frac{\sqrt{2}}{2}$

Step 1.2.2.4.2

Cancel the common factor.

$y = 2 \cdot - 1 \frac{\sqrt{2}}{2}$

Step 1.2.2.4.3

Rewrite the expression.

$y = - 1 \sqrt{2}$

Step 1.2.2.5

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

$y = - \sqrt{2}$

Step 2

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

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Step 2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 \sin (x)$ with respect to $x$ is $- 2 \frac{d}{d x} [\sin (x)]$ .

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

Step 2.2

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

$- 2 \cos (x)$

Step 2.3

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

$- 2 \cos (\frac{3}{4} \cdot π)$

Step 2.4

Simplify.

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Step 2.4.1

Combine $\frac{3}{4}$ and $π$ .

$- 2 \cos (\frac{3 π}{4})$

Step 2.4.2

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.

$- 2 (- \cos (\frac{π}{4}))$

Step 2.4.3

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

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

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

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

Step 2.4.4.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- \sqrt{2}}{2}$

Step 2.4.4.3

Cancel the common factor.

$2 \cdot - 1 \frac{- \sqrt{2}}{2}$

Step 2.4.4.4

Rewrite the expression.

$- 1 (- \sqrt{2})$

Step 2.4.5

Multiply.

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Step 2.4.5.1

Multiply $- 1$ by $- 1$ .

$1 \sqrt{2}$

Step 2.4.5.2

Multiply $\sqrt{2}$ by $1$ .

$\sqrt{2}$

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

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

Step 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite.

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

Step 3.3.1.2

Simplify terms.

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Combine $\sqrt{2}$ and $\frac{3 π}{4}$ .

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

Step 3.3.1.3

Move $3$ to the left of $\sqrt{2}$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

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Step 3.3.3.1

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

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

Step 3.3.3.2

Combine $- \sqrt{2}$ and $\frac{4}{4}$ .

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

Step 3.3.3.3

Combine the numerators over the common denominator.

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

Step 3.3.3.4

Multiply $4$ by $- 1$ .

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

Step 3.3.3.5

Factor $- 1$ out of $- 3 \sqrt{2} π$ .

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

Step 3.3.3.6

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

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

Step 3.3.3.7

Factor $- 1$ out of $- (3 \sqrt{2} π) - (4 \sqrt{2})$ .

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

Step 3.3.3.8

Rewrite $- (3 \sqrt{2} π + 4 \sqrt{2})$ as $- 1 (3 \sqrt{2} π + 4 \sqrt{2})$ .

$y = \sqrt{2} x + \frac{- 1 (3 \sqrt{2} π + 4 \sqrt{2})}{4}$

Step 3.3.3.9

Move the negative in front of the fraction.

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

Step 4