MULTIPLE CHOICE CALCULUS 1) An object moves along a line so that its position at time t is s(t) = t^4 - 6t^3 - 2t - 1.

At what time t is the acceleration of the object zero?

A. at 3 only

B. at 0 and 3 only

C. at 0 only

D. at 1 only

2) If f(x) = e^x (sin x + cos x), then f'(x) =

A. 2e^x (cos x + sin x)

B. e^x cos x

C. e^x (cos^2x - sin^2x)

D. 2e^x cos x

1) B. at 0 and 3 only

2) D. 2eˣcosx

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Algebra I

Terms/CoefficientsFactoringQuadratics

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Basic Power Rule:

f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹

Derivative Property [Addition/Subtraction]:                                                          

Derivative Rule [Product Rule]:                                                                              

Trig Derivative:

Trig Derivative:

eˣ Derivative:

Step-by-step explanation:

*Note:

Velocity is the derivative of position.

Acceleration is the derivative of velocity.

Question 1

Step 1: Define

s(t) = t⁴ - 6t³ - 2t - 1

Step 2: Differentiate

[Velocity] Basic Power Rule:                                                                            s'(t) = 4 · t⁴⁻¹ - 3 · 6t³⁻¹ - 1 · 2t¹⁻¹[Velocity] Simplify:                                                                                            v(t) = 4t³ - 18t² - 2[Acceleration] Basic Power Rule:                                                                    v'(t) = 3 · 4t³⁻¹ - 2 · 18t²⁻¹[Acceleration] Simplify:                                                                                     a(t) = 12t² - 36t

Step 3: Solve

[Acceleration] Set up:                                                                                       12t² - 36t = 0[Time] Factor:                                                                                                    12t(t - 3) = 0[Time] Identify:                                                                                                  t = 0, 3

Question 2

Step 1: Define

f(x) = eˣ(sinx + cosx)

Step 2: Differentiate

[Derivative] Product Rule:                                                                              [Derivative] Rewrite [Derivative Property - Addition]:                                  [Derivative] eˣ Derivative:                                                                               [Derivative] Trig Derivatives:                                                                         [Derivative] Factor:                                                                                         [Derivative] Combine like terms:                                                                   [Derivative] Multiply:                                                                                      

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

Unit: Derivatives

Book: College Calculus 10e

If you make it a little more clear i can answer it