Math Problem Statement

Find f if f''(x) = sin x + cos x, f'(0) = 5, and f(0) = 5.

Solution

To solve for f(x)f(x) given the conditions:

  1. f(x)=sinx+cosxf''(x) = \sin x + \cos x
  2. f(0)=5f'(0) = 5
  3. f(0)=5f(0) = 5

we'll proceed by integrating step-by-step.

Step 1: Find f(x)f'(x) by integrating f(x)f''(x).

Given: f(x)=sinx+cosxf''(x) = \sin x + \cos x Integrate f(x)f''(x) with respect to xx: f(x)=(sinx+cosx)dxf'(x) = \int (\sin x + \cos x) \, dx f(x)=cosx+sinx+C1f'(x) = -\cos x + \sin x + C_1 where C1C_1 is the constant of integration.

Step 2: Use f(0)=5f'(0) = 5 to solve for C1C_1.

Substitute x=0x = 0 into f(x)f'(x): f(0)=cos(0)+sin(0)+C1=1+0+C1=5f'(0) = -\cos(0) + \sin(0) + C_1 = -1 + 0 + C_1 = 5 C1=6C_1 = 6 So, f(x)=cosx+sinx+6f'(x) = -\cos x + \sin x + 6

Step 3: Find f(x)f(x) by integrating f(x)f'(x).

Now integrate f(x)f'(x): f(x)=(cosx+sinx+6)dxf(x) = \int (-\cos x + \sin x + 6) \, dx f(x)=sinxcosx+6x+C2f(x) = -\sin x - \cos x + 6x + C_2 where C2C_2 is another constant of integration.

Step 4: Use f(0)=5f(0) = 5 to solve for C2C_2.

Substitute x=0x = 0 into f(x)f(x): f(0)=sin(0)cos(0)+60+C2=01+0+C2=5f(0) = -\sin(0) - \cos(0) + 6 \cdot 0 + C_2 = 0 - 1 + 0 + C_2 = 5 C2=6C_2 = 6 So, f(x)=sinxcosx+6x+6f(x) = -\sin x - \cos x + 6x + 6

Final Answer

f(x)=sinxcosx+6x+6f(x) = -\sin x - \cos x + 6x + 6

Would you like more details on any step, or have any further questions?


Additional Related Questions:

  1. How would the answer change if f(0)f(0) were a different value?
  2. What if f(x)f'(x) had additional terms beyond trigonometric functions?
  3. Can we use alternative methods, like Laplace transforms, for such problems?
  4. How can we verify our solution by differentiating f(x)f(x) back?
  5. What if f(x)f(x) were required to satisfy an initial condition at a different point?

Tip:

Remember to check initial conditions carefully, as they determine the constants in integration solutions.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Integration
Initial Value Problem

Formulas

f'(x) = ∫(f''(x)) dx
f(x) = ∫(f'(x)) dx

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12