Question

If anyone can help with my Calculus that'd be great. i'm having trouble and it's due next hour

Answer 1

`\displaystyle (dA)/(dt) = 8 \pi(16 \pi + 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

Calculus

Derivatives

Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Product Rule]:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

Implicit Differentiation

Explanation:

Step 1: Define

Identify

`\displaystyle A = 2\pi rh`

`\displaystyle r = 2`

`\displaystyle h = 4`

`\displaystyle (dr)/(dt) = 16\pi`

`\displaystyle (dh)/(dt) = 2`

Step 2: Differentiate

1. [Implicit Differentiation] Product Rule:
   `\displaystyle (dA)/(dt) = 2\pi \bigg( (d)/(dt)[r]h + r(d)/(dt)[h] \bigg)`
2. Simplify:
   `\displaystyle (dA)/(dt) = 2\pi \bigg( (dr)/(dt)h + r(dh)/(dt) \bigg)`

Step 3: Solve

1. Substitute in variables [Derivative]:
   `\displaystyle (dA)/(dt) = 2\pi \bigg( 16\pi(4) + 2(2) \bigg)`
2. Evaluate:
   `\displaystyle (dA)/(dt) = 8 \pi(16 \pi + 1)`

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

Unit: Implicit Differentiation

Book: College Calculus 10e