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liberstina [14]
3 years ago
10

[4] 7. Find f if f"(x) = x^2+ sin x f'(0) = 2 f(0) =4

Mathematics
1 answer:
sp2606 [1]3 years ago
4 0

Answer:

f(x) = \frac{x^4}{12} - sin(x) + 3x + 4

General Formulas and Concepts:

<u>Calculus</u>

  • Antiderivatives
  • Integration Constant C
  • [Int Rule] Reverse Power Rule: \int {x^n} \, dx = \frac{x^{n+1}}{n+1} + C
  • Integration Property 1: \int {cf(x)} \, dx = c\int {f(x)} \, dx
  • Integration Property 2: \int {f(x)+g(x)} \, dx = \int {f(x)} \, dx + \int {g(x)} \, dx

Step-by-step explanation:

<u>Step 1: Define</u>

f"(x) = x² + sin(x)

Condition f'(0) = 2

Condition f(0) = 4

<u>Step 2: Integrate Pt. 1</u>

  1. Set up:                                                                                                   f'(x) = \int {f"(x)} \, dx
  2. Substitute:                                                                                            f'(x) = \int [{x^2 + sin(x)}] \, dx
  3. Rewrite [Int Property 2]:                                                                          f'(x) = \int {x^2} \, dx + \int {sin(x)} \, dx
  4. Integrate [Reverse Power Rule/Trig]:                                                    f'(x) = \frac{x^3}{3}  - cos(x) + C

<u>Step 3: Find f'(x)</u>

<em>Use the given condition to find the differential equation.</em>

  1. Substitute:                    f'(0) = \frac{0^3}{3}  - cos(0) + C
  2. Substitute:                    2 = \frac{0^3}{3}  - cos(0) + C
  3. Evaluate:                      2 = 0 - 1 + C
  4. Solve:                           3 = C
  5. Define:                         f'(x) = \frac{x^3}{3}  - cos(x) + 3

<u>Step 4: Integrate Pt. 2</u>

  1. Set up:                                                                                                   f(x) = \int {f'(x)} \, dx
  2. Substitute:                                                                                              f(x) = \int [{\frac{x^3}{3}  - cos(x) + 3}] \, dx
  3. Rewrite [Int Property 2]:                                                                                f(x) = \int {\frac{x^3}{3} } \, dx + \int {-cos(x)} \, dx  + \int {3} \, dx
  4. Rewrite [Int Property 1]:                                                                            f(x) = \frac{1}{3} \int {x^3} \, dx - \int {cos(x)} \, dx  + \int {3} \, dx
  5. Integrate {Reverse Power Rule/Trig]:                                                      f(x) = \frac{1}{3}(\frac{x^4}{4} ) - sin(x) + 3x + C
  6. Simplify:                                                                                                    f(x) = \frac{x^4}{12} - sin(x) + 3x + C

<u>Step 5: Find f(x)</u>

<em>Use the given condition to find the equation.</em>

  1. Substitute:                    f(0) = \frac{0^4}{12} - sin(0) + 3(0) + C
  2. Substitute:                    4 = \frac{0^4}{12} - sin(0) + 3(0) + C
  3. Evaluate:                       4 = 0 - 0 + 0+ C
  4. Solve:                            4 = C
  5. Define:                          f(x) = \frac{x^4}{12} - sin(x) + 3x + 4
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