Question

The curve


`y = √(x - 3)`
has a tangent with gradient

`(1)/(2)`
at point N.
Find the coordinates of N.​

Answer 1

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) = (1)/(b^m)`
• Exponential Rule [Root Rewrite]:
  `\displaystyle \sqrt[n]{x} = x^{(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 (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Explanation:

Step 1: Define

`\displaystyle y = √(x - 3)`

`\displaystyle y' = (1)/(2)`

Step 2: Differentiate

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

Step 3: Solve

Find coordinates

x-coordinate

1. Substitute in y' [Derivative]:
   `\displaystyle (1)/(2) = (1)/(2√(x - 3))`
2. [Multiplication Property of Equality] Multiply 2 on both sides:
   `\displaystyle 1 = (1)/(√(x - 3))`
3. [Multiplication Property of Equality] Multiply √(x - 3) on both sides:
   `\displaystyle √(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 = √(4 - 3)`
2. [√Radical] Subtract:
   `\displaystyle y = √(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