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
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
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
- [Derivative] Rewrite [Leibniz's Notation]: [tex]\displaystyle ds = (5t^3 + \frac{9}{t^2})dt[/tex]
- [Equality Property] Integrate both sides: [tex]\displaystyle \int {} \, ds = \int {(5t^3 + \frac{9}{t^2})} \, dt[/tex]
- [Left Integral] Reverse Power Rule: [tex]\displaystyle s = \int {(5t^3 + \frac{9}{t^2})} \, dt[/tex]
- [Right Integral] Rewrite [Integration Property - Addition]: [tex]\displaystyle s = \int {5t^3} \, dt + \int {\frac{9}{t^2}} \, dt[/tex]
- [Right Integrals] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle s = 5\int {t^3} \, dt + 9\int {\frac{1}{t^2}} \, dt[/tex]
- [Right Integrals] Rewrite [Exponential Rule - Rewrite]: [tex]\displaystyle s = 5\int {t^3} \, dt + 9\int {t^{-2}} \, dt[/tex]
- [Right Integrals] Reverse Power Rule: [tex]\displaystyle s = 5(\frac{t^4}{4}) + 9(\frac{t^{-1}}{-1}) + C[/tex]
- [Right Integrals] Rewrite [Exponential Rule - Rewrite]: [tex]\displaystyle s = 5(\frac{t^4}{4}) + 9(\frac{1}{t}) + C[/tex]
- Multiply: [tex]\displaystyle s = \frac{5t^4}{4} + \frac{9}{t} + C[/tex]
Step 3: Solve
- Substitute in variables: [tex]\displaystyle 8 = \frac{5(1)^4}{4} + \frac{9}{1} + C[/tex]
- Evaluate exponents: [tex]\displaystyle 8 = \frac{5}{4} + \frac{9}{1} + C[/tex]
- Divide: [tex]\displaystyle 8 = \frac{5}{4} + 9 + C[/tex]
- Add: [tex]\displaystyle 8 = \frac{41}{4} + C[/tex]
- [Subtraction Property of Equality] Isolate C: [tex]\displaystyle \frac{-9}{4} = C[/tex]
- 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
