Answer :

Space

Answer:

[tex]\displaystyle s = \frac{5t^4}{4} + \frac{9}{t} - \frac{9}{4}[/tex]

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

  • Exponential Rule [Rewrite]:                                                                           [tex]\displaystyle b^{-m} = \frac{1}{b^m}[/tex]

Calculus

Derivatives

Derivative Notation

Solving Differentials - Integrals

Integration Constant C

Integration Rule [Reverse Power Rule]:                                                               [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Property [Multiplied Constant]:                                                         [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

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:

Ignore the Integration Constant C on the left hand side of the differential equation when integrating.

Step 1: Define

[tex]\displaystyle \frac{ds}{dt} = 5t^3 + \frac{9}{t^2}[/tex]

t = 1

s = 8

Step 2: Integrate

  1. [Derivative] Rewrite [Leibniz's Notation]:                                                     [tex]\displaystyle ds = (5t^3 + \frac{9}{t^2})dt[/tex]
  2. [Equality Property] Integrate both sides:                                                     [tex]\displaystyle \int {} \, ds = \int {(5t^3 + \frac{9}{t^2})} \, dt[/tex]
  3. [Left Integral] Reverse Power Rule:                                                             [tex]\displaystyle s = \int {(5t^3 + \frac{9}{t^2})} \, dt[/tex]
  4. [Right Integral] Rewrite [Integration Property - Addition]:                           [tex]\displaystyle s = \int {5t^3} \, dt + \int {\frac{9}{t^2}} \, dt[/tex]
  5. [Right Integrals] Rewrite [Integration Property - Multiplied Constant]:     [tex]\displaystyle s = 5\int {t^3} \, dt + 9\int {\frac{1}{t^2}} \, dt[/tex]
  6. [Right Integrals] Rewrite [Exponential Rule - Rewrite]:                               [tex]\displaystyle s = 5\int {t^3} \, dt + 9\int {t^{-2}} \, dt[/tex]
  7. [Right Integrals] Reverse Power Rule:                                                         [tex]\displaystyle s = 5(\frac{t^4}{4}) + 9(\frac{t^{-1}}{-1}) + C[/tex]
  8. [Right Integrals] Rewrite [Exponential Rule - Rewrite]:                               [tex]\displaystyle s = 5(\frac{t^4}{4}) + 9(\frac{1}{t}) + C[/tex]
  9. Multiply:                                                                                                         [tex]\displaystyle s = \frac{5t^4}{4} + \frac{9}{t} + C[/tex]

Step 3: Solve

  1. Substitute in variables:                                                                                 [tex]\displaystyle 8 = \frac{5(1)^4}{4} + \frac{9}{1} + C[/tex]
  2. Evaluate exponents:                                                                                     [tex]\displaystyle 8 = \frac{5}{4} + \frac{9}{1} + C[/tex]
  3. Divide:                                                                                                           [tex]\displaystyle 8 = \frac{5}{4} + 9 + C[/tex]
  4. Add:                                                                                                               [tex]\displaystyle 8 = \frac{41}{4} + C[/tex]
  5. [Subtraction Property of Equality] Isolate C:                                               [tex]\displaystyle \frac{-9}{4} = C[/tex]
  6. Rewrite:                                                                                                          [tex]\displaystyle C = \frac{-9}{4}[/tex]

Particular Solution: [tex]\displaystyle s = \frac{5t^4}{4} + \frac{9}{t} - \frac{9}{4}[/tex]

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

Unit: Differentials Equations and Slope Fields

Book: College Calculus 10e

Other Questions