Calculus Examples

$f (t) = t^{- 3}$ ;, $(\frac{1}{2}, 8)$

Step 1

Find the first derivative and evaluate at $t = \frac{1}{2}$ and $y = 8$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = - 3$ .

$- 3 t^{- 4}$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 3 \frac{1}{t^{4}}$

Step 1.2.2

Combine $- 3$ and $\frac{1}{t^{4}}$ .

$\frac{- 3}{t^{4}}$

Step 1.2.3

Move the negative in front of the fraction.

$- \frac{3}{t^{4}}$

Step 1.3

Evaluate the derivative at $t = \frac{1}{2}$ .

$- \frac{3}{{(\frac{1}{2})}^{4}}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the denominator.

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Step 1.4.1.1

Apply the product rule to $\frac{1}{2}$ .

$- \frac{3}{\frac{1^{4}}{2^{4}}}$

Step 1.4.1.2

One to any power is one.

$- \frac{3}{\frac{1}{2^{4}}}$

Step 1.4.1.3

Raise $2$ to the power of $4$ .

$- \frac{3}{\frac{1}{16}}$

Step 1.4.2

Multiply the numerator by the reciprocal of the denominator.

$- (3 \cdot 16)$

Step 1.4.3

Multiply $- (3 \cdot 16)$ .

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Step 1.4.3.1

Multiply $3$ by $16$ .

$- 1 \cdot 48$

Step 1.4.3.2

Multiply $- 1$ by $48$ .

$- 48$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $- 48$ and a given point $(\frac{1}{2}, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = - 48 \cdot (x - (\frac{1}{2}))$

Step 2.2

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

$y - 8 = - 48 \cdot (x - \frac{1}{2})$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 48 \cdot (x - \frac{1}{2})$ .

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Step 2.3.1.1

Rewrite.

$y - 8 = 0 + 0 - 48 \cdot (x - \frac{1}{2})$

Step 2.3.1.2

Simplify by adding zeros.

$y - 8 = - 48 \cdot (x - \frac{1}{2})$

Step 2.3.1.3

Apply the distributive property.

$y - 8 = - 48 x - 48 (- \frac{1}{2})$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y - 8 = - 48 x - 48 (\frac{- 1}{2})$

Step 2.3.1.4.2

Factor $2$ out of $- 48$ .

$y - 8 = - 48 x + 2 (- 24) \frac{- 1}{2}$

Step 2.3.1.4.3

Cancel the common factor.

$y - 8 = - 48 x + 2 \cdot - 24 \frac{- 1}{2}$

Step 2.3.1.4.4

Rewrite the expression.

$y - 8 = - 48 x - 24 \cdot - 1$

Step 2.3.1.5

Multiply $- 24$ by $- 1$ .

$y - 8 = - 48 x + 24$

Step 2.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.3.2.1

Add $8$ to both sides of the equation.

$y = - 48 x + 24 + 8$

Step 2.3.2.2

Add $24$ and $8$ .

$y = - 48 x + 32$

Step 3