Calculus Examples

$y = \cos (x)$ at $x = \frac{π}{2}$

Step 1

Find the corresponding $y$ -value to $x = \frac{π}{2}$ .

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Step 1.1

Substitute $\frac{π}{2}$ in for $x$ .

$y = \cos (\frac{π}{2})$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \cos (\frac{π}{2})$

Step 1.2.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$y = 0$

Step 2

Find the first derivative and evaluate at $x = \frac{π}{2}$ and $y = 0$ to find the slope of the tangent line.

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Step 2.1

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

$- \sin (x)$

Step 2.2

Evaluate the derivative at $x = \frac{π}{2}$ .

$- \sin (\frac{π}{2})$

Step 2.3

Simplify.

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Step 2.3.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$- 1 \cdot 1$

Step 2.3.2

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 $(\frac{π}{2}, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = - 1 \cdot (x - (\frac{π}{2}))$

Step 3.2

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

$y + 0 = - 1 \cdot (x - \frac{π}{2})$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Add $y$ and $0$ .

$y = - 1 \cdot (x - \frac{π}{2})$

Step 3.3.2

Simplify $- 1 \cdot (x - \frac{π}{2})$ .

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Step 3.3.2.1

Apply the distributive property.

$y = - 1 x - 1 (- \frac{π}{2})$

Step 3.3.2.2

Rewrite $- 1 x$ as $- x$ .

$y = - x - 1 (- \frac{π}{2})$

Step 3.3.2.3

Multiply $- 1 (- \frac{π}{2})$ .

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Step 3.3.2.3.1

Multiply $- 1$ by $- 1$ .

$y = - x + 1 \frac{π}{2}$

Step 3.3.2.3.2

Multiply $\frac{π}{2}$ by $1$ .

$y = - x + \frac{π}{2}$

Step 4