ThomasSu
Answered

The curve
[tex]y = \sqrt{x - 3} [/tex]
has a tangent with gradient
[tex] \frac{1}{2} [/tex]
at point N.
Find the coordinates of N.​

Answer :

Space

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]:                                                                              [tex]\displaystyle b^{-m} = \frac{1}{b^m}[/tex]
  • Exponential Rule [Root Rewrite]:                                                                     [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]

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]:                                                                                    [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]

Step-by-step explanation:

Step 1: Define

[tex]\displaystyle y = \sqrt{x - 3}[/tex]

[tex]\displaystyle y' = \frac{1}{2}[/tex]

Step 2: Differentiate

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

Step 3: Solve

Find coordinates

x-coordinate

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

y-coordinate

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

∴ Coordinates of Point N is (4, 1).

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

Unit: Derivatives

Book: College Calculus 10e

Other Questions