CALCULUS: Find the velocity, v(t), for an object moving along the x-axis if the acceleration, a(t), is a(t) = cos(t) - sin(t) and v(0) = 3. A. v(t) = sin(t) + cos(t) + 3
B. v(t) = sin(t) + cos(t) + 2
C. v(t) = sin(t) - cos(t) + 3
D. v(t) = sin(t) - cos(t) + 4

I picked A., which was marked wrong, although I truly believe that is the answer because I got sin(t) - (-cos(t)) + 3. Please tell me whether you believe this is right or am I doing something wrong. I really want to learn/master this concept. Thanks is advance.

B. v(t) = sin(t) + cos(t) + 2

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right  

Equality Properties

Multiplication Property of EqualityDivision Property of EqualityAddition Property of EqualitySubtraction Property of Equality

Algebra I

Function Notation

Calculus

Antiderivatives - Integrals

Integration Constant C

Solving Integration Equations

Integration Property [Addition/Subtraction]:                                                        

Step-by-step explanation:

*Note: Remember that the velocity function is the integral of the acceleration function/acceleration is the derivative of velocity.

Step 1: Define

a(t) = cos(t) - sin(t)

v(0) = 3

Step 2: Integrate

Set up integral:                                                                                               [integral] Rewrite [Integration Property - Subtraction]:                                 [Integral] Trig integration:                                                                              [Velocity Integration] Simplify:                                                                          

Step 3: Find Function

We need to solve for the entire function, meaning we need to find constant C.

Substitute in given point [Velocity Integration]:                                           [Velocity Integration] Substitute:                                                                   [Velocity Integration] Evaluate trig:                                                               [Velocity Integration] Add:                                                                             [Velocity Integration] Isolate C [Subtraction Property of Equality]:              [Velocity Integration] Rewrite:                                                                       [Velocity Function] Substitute in C [Velocity Integration]:                          

Topic: Calculus

Unit: Basic Integration

Book: College Calculus 10e

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