Question

The curve

Answer 1

Answer:

Point N(4, 1)

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right  

Equality Properties

• Multiplication Property of Equality
• Division Property of Equality
• Addition Property of Equality
• Subtraction Property of Equality

Algebra I

• Coordinates (x, y)
• Functions
• Function Notation
• Terms/Coefficients
• Anything to the 0th power is 1
• Exponential Rule [Rewrite]:                                                                              $\displaystyle b^{-m} = \frac{1}{b^m}$
• Exponential Rule [Root Rewrite]:                                                                     $\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}$

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:                                                                                    $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

Step 1: Define

$\displaystyle y = \sqrt{x - 3}$

$\displaystyle y' = \frac{1}{2}$

Step 2: Differentiate

1. [Function] Rewrite [Exponential Rule - Root Rewrite]:                                   $\displaystyle y = (x - 3)^{\frac{1}{2}}$
2. Chain Rule:                                                                                                        $\displaystyle y' = \frac{d}{dx}[(x - 3)^{\frac{1}{2}}] \cdot \frac{d}{dx}[x - 3]$
3. Basic Power Rule:                                                                                             $\displaystyle y' = \frac{1}{2}(x - 3)^{\frac{1}{2} - 1} \cdot (1 \cdot x^{1 - 1} - 0)$
4. Simplify:                                                                                                             $\displaystyle y' = \frac{1}{2}(x - 3)^{-\frac{1}{2}} \cdot 1$
5. Multiply:                                                                                                             $\displaystyle y' = \frac{1}{2}(x - 3)^{-\frac{1}{2}}$
6. [Derivative] Rewrite [Exponential Rule - Rewrite]:                                          $\displaystyle y' = \frac{1}{2(x - 3)^{\frac{1}{2}}}$
7. [Derivative] Rewrite [Exponential Rule - Root Rewrite]:                                 $\displaystyle y' = \frac{1}{2\sqrt{x - 3}}$

Step 3: Solve

Find coordinates

x-coordinate

1. Substitute in y' [Derivative]:                                                                             $\displaystyle \frac{1}{2} = \frac{1}{2\sqrt{x - 3}}$
2. [Multiplication Property of Equality] Multiply 2 on both sides:                      $\displaystyle 1 = \frac{1}{\sqrt{x - 3}}$
3. [Multiplication Property of Equality] Multiply √(x - 3) on both sides:            $\displaystyle \sqrt{x - 3} = 1$
4. [Equality Property] Square both sides:                                                           $\displaystyle x - 3 = 1$
5. [Addition Property of Equality] Add 3 on both sides:                                    $\displaystyle x = 4$

y-coordinate

1. Substitute in x [Function]:                                                                                $\displaystyle y = \sqrt{4 - 3}$
2. [√Radical] Subtract:                                                                                          $\displaystyle y = \sqrt{1}$
3. [√Radical] Evaluate:                                                                                         $\displaystyle y = 1$

∴ Coordinates of Point N is (4, 1).

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Derivatives

Book: College Calculus 10e