Question

AP CALC AB QUESTION PLEASE HELP AND PROVIDE AN EXPLANATION

Answer 1

Answer:

False

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$  

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

Explanation:

Step 1: Define

Identify

$\displaystyle \int\limits^3_2 {\frac{1}{3x - 2}} \, dx \stackrel{?}{=} \int\limits^7_4 {\frac{1}{u}} \, du$

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]:                 $\displaystyle \int\limits^3_2 {\frac{1}{3x - 2}} \, dx = \frac{1}{3} \int\limits^3_2 {\frac{3}{3x - 2}} \, dx$
2. [Integral] U-Substitution:                                                                               $\displaystyle \int\limits^3_2 {\frac{1}{3x - 2}} \, dx = \frac{1}{3} \int\limits^7_4 {\frac{1}{u}} \, du$
3. Compare:                                                                                                       $\displaystyle \int\limits^3_2 {\frac{1}{3x - 2}} \, dx = \frac{1}{3} \int\limits^7_4 {\frac{1}{u}} \, du \neq \int\limits^7_4 {\frac{1}{u}} \, du$

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

Unit: Integration