Calculus Examples

$f (x) = 0.5 x^{2} - 2 x - 4$ , $x = 3$

Step 1

Find the corresponding $y$ -value to $x = 3$ .

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Step 1.1

Substitute $3$ in for $x$ .

$y = 0.5 {(3)}^{2} - 2 \cdot 3 - 4$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 0.5 \cdot 3^{2} - 2 \cdot 3 - 4$

Step 1.2.2

Remove parentheses.

$y = 0.5 {(3)}^{2} - 2 \cdot 3 - 4$

Step 1.2.3

Simplify $0.5 {(3)}^{2} - 2 \cdot 3 - 4$ .

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Raise $3$ to the power of $2$ .

$y = 0.5 \cdot 9 - 2 \cdot 3 - 4$

Step 1.2.3.1.2

Multiply $0.5$ by $9$ .

$y = 4.5 - 2 \cdot 3 - 4$

Step 1.2.3.1.3

Multiply $- 2$ by $3$ .

$y = 4.5 - 6 - 4$

Step 1.2.3.2

Simplify by subtracting numbers.

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Step 1.2.3.2.1

Subtract $6$ from $4.5$ .

$y = - 1.5 - 4$

Step 1.2.3.2.2

Subtract $4$ from $- 1.5$ .

$y = - 5.5$

Step 2

Find the first derivative and evaluate at $x = 3$ and $y = - 5.5$ to find the slope of the tangent line.

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Step 2.1

By the Sum Rule, the derivative of $0.5 x^{2} - 2 x - 4$ with respect to $x$ is $\frac{d}{d x} [0.5 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [0.5 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 4]$

Step 2.2

Evaluate $\frac{d}{d x} [0.5 x^{2}]$ .

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Step 2.2.1

Since $0.5$ is constant with respect to $x$ , the derivative of $0.5 x^{2}$ with respect to $x$ is $0.5 \frac{d}{d x} [x^{2}]$ .

$0.5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 4]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0.5 (2 x) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 4]$

Step 2.2.3

Multiply $2$ by $0.5$ .

$1 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 4]$

Step 2.2.4

Multiply $x$ by $1$ .

$x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 4]$

Step 2.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 2.3.1

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

$x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x - 2 \cdot 1 + \frac{d}{d x} [- 4]$

Step 2.3.3

Multiply $- 2$ by $1$ .

$x - 2 + \frac{d}{d x} [- 4]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$x - 2 + 0$

Step 2.4.2

Add $x - 2$ and $0$ .

$x - 2$

Step 2.5

Evaluate the derivative at $x = 3$ .

$(3) - 2$

Step 2.6

Subtract $2$ from $3$ .

$1$

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 $(3, - 5.5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 5.5) = 1 \cdot (x - (3))$

Step 3.2

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

$y + 5.5 = 1 \cdot (x - 3)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Multiply $x - 3$ by $1$ .

$y + 5.5 = x - 3$

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 $5.5$ from both sides of the equation.

$y = x - 3 - 5.5$

Step 3.3.2.2

Subtract $5.5$ from $- 3$ .

$y = x - 8.5$

Step 4