Respuesta :

Space

Answer:

Part A:  [tex]\displaystyle f(x) = \frac{19}{2}x^2 + 15x + C[/tex]

Part B:  [tex]\displaystyle f(x) = \frac{19}{2}x^2 + 15x - 5[/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

Functions

  • Function Notation

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Differential Equations

Integration

  • Integrals
  • [Indefinite 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:

Step 1: Define

Identify

[tex]\displaystyle f'(x) = 19x + 15[/tex]

Step 2: Find Antiderivative

  1. [Derivative] Integrate both sides:                                                                 [tex]\displaystyle \int {f'(x)} \, dx = \int {19x + 15} \, dx[/tex]
  2. [Left Integral] Simplify:                                                                                   [tex]\displaystyle f(x) = \int {19x + 15} \, dx[/tex]
  3. [Integral] Rewrite [Integration Property - Addition/Subtraction]:               [tex]\displaystyle f(x) = \int {19x} \, dx + \int {15} \, dx[/tex]
  4. [Integrals] Rewrite [Integration Property - Multiplied Constant]:               [tex]\displaystyle f(x) = 19 \int {x} \, dx + 15 \int {} \, dx[/tex]
  5. [Integrals] Integration Rule [Reverse Power Rule]:                                     [tex]\displaystyle f(x) = 19 \bigg( \frac{x^2}{2} \bigg) + 15x + C[/tex]
  6. Simplify:                                                                                                         [tex]\displaystyle f(x) = \frac{19}{2}x^2 + 15x + C[/tex]

Step 3: Find Particular Solution

  1. Substitute in function value [Function f(x)]:                                                 [tex]\displaystyle 87 = \frac{19}{2}(-4)^2 + 15(-4) + C[/tex]
  2. Evaluate:                                                                                                         [tex]\displaystyle 87 = 92 + C[/tex]
  3. Solve:                                                                                                             [tex]\displaystyle C = -5[/tex]
  4. Substitute in C [General Solution]:                                                               [tex]\displaystyle f(x) = \frac{19}{2}x^2 + 15x - 5[/tex]

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

Unit: Differential Equations

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