Calculus Examples

$y = 2 - \sin (x)$ at $x = π$

Step 1

Find the corresponding $y$ -value to $x = π$ .

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Step 1.1

Substitute $π$ in for $x$ .

$y = 2 - \sin (π)$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 2 - \sin (π)$

Step 1.2.2

Simplify $2 - \sin (π)$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

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

$y = 2 - \sin (0)$

Step 1.2.2.1.2

The exact value of $\sin (0)$ is $0$ .

$y = 2 - 0$

Step 1.2.2.1.3

Multiply $- 1$ by $0$ .

$y = 2 + 0$

Step 1.2.2.2

Add $2$ and $0$ .

$y = 2$

Step 2

Find the first derivative and evaluate at $x = π$ and $y = 2$ 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 $2 - \sin (x)$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- \sin (x)]$ .

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

$0 - \cos (x)$

Step 2.3

Subtract $\cos (x)$ from $0$ .

$- \cos (x)$

Step 2.4

Evaluate the derivative at $x = π$ .

$- \cos (π)$

Step 2.5

Simplify.

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Step 2.5.1

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.

$- - \cos (0)$

Step 2.5.2

The exact value of $\cos (0)$ is $1$ .

$- (- 1 \cdot 1)$

Step 2.5.3

Multiply $- (- 1 \cdot 1)$ .

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Step 2.5.3.1

Multiply $- 1$ by $1$ .

$- - 1$

Step 2.5.3.2

Multiply $- 1$ by $- 1$ .

$1$

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

$y - (2) = 1 \cdot (x - (π))$

Step 3.2

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

$y - 2 = 1 \cdot (x - π)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Multiply $x - π$ by $1$ .

$y - 2 = x - π$

Step 3.3.2

Add $2$ to both sides of the equation.

$y = x - π + 2$

Step 4