Question

AP CALC AB QUESTION PLEASE HELP AND PROVIDE AN EXPLANATION

Answer 1

False

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals
• Integral Notation

Integration Property [Multiplied Constant]:
`\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx`

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

`\displaystyle \int\limits^3_2 {(1)/(3x - 2)} \, dx \stackrel{?}{=} \int\limits^7_4 {(1)/(u)} \, du`

Step 2: Verify Pt. 1

Identify variables for u-substitution.

1. Set u:
   `\displaystyle u = 3x - 2`
2. [u] Differentiate [Derivative Properties]:
   `\displaystyle du = 3 \ dx`
3. [Limits] Switch:
   `\displaystyle \left \{ {{x = 3 ,\ u = 3(3) - 2 = 7} \atop {x = 2 ,\ u = 3(2) - 2 = 4}} \right.`

Step 3: Verify Pt. 2

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int\limits^3_2 {(1)/(3x - 2)} \, dx = (1)/(3) \int\limits^3_2 {(3)/(3x - 2)} \, dx`
2. [Integral] U-Substitution:
   `\displaystyle \int\limits^3_2 {(1)/(3x - 2)} \, dx = (1)/(3) \int\limits^7_4 {(1)/(u)} \, du`
3. Compare:
   `\displaystyle \int\limits^3_2 {(1)/(3x - 2)} \, dx = (1)/(3) \int\limits^7_4 {(1)/(u)} \, du \\eq \int\limits^7_4 {(1)/(u)} \, du`

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

Unit: Integration