Question

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.

Answer 1

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]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

Explanation:

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

`\displaystyle v'(t) = a(t)\\v(t) = \int {a(t)} \, dt`

Step 1: Define

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

v(0) = 3

Step 2: Integrate

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

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

Topic: Calculus

Unit: Basic Integration

Book: College Calculus 10e