Math  /  Calculus

QuestionFind a function f(x)f(x) that satisfies f(x)=x48xf^{\prime}(x)=x^{4}-8 x and f(1)=2f(1)=2. Answer:

Studdy Solution

STEP 1

Assumptions
1. We are given the derivative of the function, f(x)=x48x f^{\prime}(x) = x^4 - 8x .
2. We need to find the function f(x) f(x) such that it satisfies the given derivative.
3. We are also given an initial condition, f(1)=2 f(1) = 2 .

STEP 2

To find the function f(x) f(x) , we need to integrate the derivative f(x)=x48x f^{\prime}(x) = x^4 - 8x .

STEP 3

Set up the integral of the derivative to find f(x) f(x) .
f(x)=(x48x)dx f(x) = \int (x^4 - 8x) \, dx

STEP 4

Integrate each term separately.
f(x)=x4dx8xdx f(x) = \int x^4 \, dx - \int 8x \, dx

STEP 5

Calculate the integral of x4 x^4 .
x4dx=x55+C1 \int x^4 \, dx = \frac{x^5}{5} + C_1

STEP 6

Calculate the integral of 8x 8x .
8xdx=8x22+C2=4x2+C2 \int 8x \, dx = 8 \cdot \frac{x^2}{2} + C_2 = 4x^2 + C_2

STEP 7

Combine the results of the integrals to find f(x) f(x) .
f(x)=x554x2+C f(x) = \frac{x^5}{5} - 4x^2 + C
where C=C1C2 C = C_1 - C_2 is the constant of integration.

STEP 8

Use the initial condition f(1)=2 f(1) = 2 to solve for C C .
f(1)=155412+C=2 f(1) = \frac{1^5}{5} - 4 \cdot 1^2 + C = 2

STEP 9

Simplify the equation using the initial condition.
154+C=2 \frac{1}{5} - 4 + C = 2

STEP 10

Solve for C C .
C=2+415 C = 2 + 4 - \frac{1}{5}
C=615 C = 6 - \frac{1}{5}
C=30515 C = \frac{30}{5} - \frac{1}{5}
C=295 C = \frac{29}{5}

STEP 11

Substitute the value of C C back into the expression for f(x) f(x) .
f(x)=x554x2+295 f(x) = \frac{x^5}{5} - 4x^2 + \frac{29}{5}
This is the function f(x) f(x) that satisfies the given conditions.

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