Mathematics, 20.12.2020 01:50 Kana81

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

Answers: 2

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Answer from: dontcareanyonemo

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

General Formulas and Concepts:

Calculus

AntiderivativesIntegration 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:

Step 1: Define

f"(x) = x² + sin(x)

Condition f'(0) = 2

Condition f(0) = 4

Step 2: Integrate Pt. 1

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

Step 3: Find f'(x)

Use the given condition to find the differential equation.

Substitute: $f'(0) = \frac{0^3}{3} - cos(0) + C$ Substitute: $2 = \frac{0^3}{3} - cos(0) + C$ Evaluate: 2 = 0 - 1 + C

Solve:

Define: $f'(x) = \frac{x^3}{3} - cos(x) + 3$

Step 4: Integrate Pt. 2

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

Step 5: Find f(x)

Use the given condition to find the equation.

Substitute: $f(0) = \frac{0^4}{12} - sin(0) + 3(0) + C$ Substitute: $4 = \frac{0^4}{12} - sin(0) + 3(0) + C$ Evaluate: 4 = 0 - 0 + 0+ C

Solve:

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

Answer from: Quest

answer: plz put a pic in so i can answer it

step-by-step explanation:

Answer from: Quest

step-by-step explanation: sorry if not correct becaude my brother is loud and i could not think!

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

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