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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.

CALCULUS Find the velocity vt for an object moving along the xaxis if the acceleration at is at cost sint and v0 3 A vt sint cost 3 B vt sint cost 2 C vt sint c class=

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Answer:

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

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  

Equality Properties

  • Multiplication Property of Equality
  • Division Property of Equality
  • Addition Property of Equality
  • Subtraction Property of Equality

Algebra I

  • Function Notation

Calculus

Antiderivatives - Integrals

Integration Constant C

Solving Integration Equations

Integration Property [Addition/Subtraction]:                                                         [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Step-by-step explanation:

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

[tex]\displaystyle v'(t) = a(t)\\v(t) = \int {a(t)} \, dt[/tex]

Step 1: Define

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

v(0) = 3

Step 2: Integrate

  1. Set up integral:                                                                                               [tex]\displaystyle v(t) = \int {cos(t) - sin(t)} \, dt[/tex]
  2. [integral] Rewrite [Integration Property - Subtraction]:                                 [tex]\displaystyle v(t) = \int {cos(t)} \, dt - \int {sin(t)} \, dt[/tex]
  3. [Integral] Trig integration:                                                                              [tex]\displaystyle v(t) = sin(t) - [-cos(t)] + C[/tex]
  4. [Velocity Integration] Simplify:                                                                          [tex]\displaystyle v(t) = sin(t) + cos(t) + C[/tex]

Step 3: Find Function

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

  1. Substitute in given point [Velocity Integration]:                                           [tex]\displaystyle v(0) = sin(0) + cos(0) + C[/tex]
  2. [Velocity Integration] Substitute:                                                                   [tex]\displaystyle 3 = sin(0) + cos(0) + C[/tex]
  3. [Velocity Integration] Evaluate trig:                                                               [tex]\displaystyle 3 = 0 + 1 + C[/tex]
  4. [Velocity Integration] Add:                                                                             [tex]\displaystyle 3 = 1 + C[/tex]
  5. [Velocity Integration] Isolate C [Subtraction Property of Equality]:              [tex]\displaystyle 2 = C[/tex]
  6. [Velocity Integration] Rewrite:                                                                       [tex]\displaystyle C = 2[/tex]
  7. [Velocity Function] Substitute in C [Velocity Integration]:                           [tex]\displaystyle v(t) = sin(t) + cos(t) + 2[/tex]

Topic: Calculus

Unit: Basic Integration

Book: College Calculus 10e

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