Find f.
f ''(x) = 20x3 + 12x2 + 10, f(0) = 9, f(1) = 12
To find the function f(x), we can integrate the second derivative of f(x) with respect to x twice.
Step 1: Integrate the second derivative to find the first derivative of f(x):
∫[f ''(x)] dx = ∫(20x^3 + 12x^2 + 10) dx
f '(x) = 5x^4 + 4x^3 + 10x + C₁
Step 2: Integrate the first derivative to find f(x):
∫[f '(x)] dx = ∫(5x^4 + 4x^3 + 10x + C₁) dx
f(x) = (5/5)x^5 + (4/4)x^4 + (10/3)x^2 + C₁x + C₂
Step 3: Use the given initial conditions to solve for C₁ and C₂:
f(0) = 9, so plugging in x = 0: 9 = (0) + (0) + (10/3)(0) + C₂
Therefore, C₂ = 9.
f(1) = 12, so plugging in x = 1: 12 = (5/5)(1)^5 + (4/4)(1)^4 + (10/3)(1)^2 + C₁(1) + 9
Simplifying this equation gives: 12 = 1 + 1 + (10/3) + C₁ + 9
C₁ = 12 - 1 - 1 - (10/3) - 9
C₁ = -9/3 = -3.
Step 4: Substitute the values of C₁ and C₂ into the equation for f(x):
f(x) = (5/5)x^5 + (4/4)x^4 + (10/3)x^2 - 3x + 9
Therefore, the function f(x) is:
f(x) = x^5 + x^4 + (10/3)x^2 - 3x + 9.
integrating once to get
f'(x) = 5x^4 + 4x^3 + 10x + c
check by taking the derivative,
integrate once more to get
f(x) = x^5 + x^4 + 5x^2 + cx + k
now use the two given points:
for f(0) =9
9 = 0+0+0+0+k , so k = 9
for (1,12)
12 = 1+1+5+c+9
c = ...
f(x) = ....
To find the function f(x), we need to integrate the given second derivative of f(x) with respect to x. Integrating twice will give us the original function f(x).
Step 1: Integrate the second derivative
Given:
f ''(x) = 20x^3 + 12x^2 + 10
Integrating f ''(x) once will give us the first derivative f '(x):
f '(x) = ∫(20x^3 + 12x^2 + 10)dx
To solve this integral, we need to apply the power rule of integration:
∫x^n dx = (1/(n+1))x^(n+1) + C, where C is the constant of integration.
Using the power rule, we integrate each term of the function f ''(x):
∫(20x^3 + 12x^2 + 10)dx = (1/(4+1))20x^(3+1) + (1/(2+1))12x^(2+1) + 10x + C1
Simplifying the integral, we get:
f '(x) = 4x^4 + 4x^3 + 10x^2 + C1
Step 2: Integrate the first derivative
Now we integrate f '(x) to find the original function f(x):
f(x) = ∫(4x^4 + 4x^3 + 10x^2 + C1)dx
Using the power rule, we integrate each term of f '(x):
∫(4x^4 + 4x^3 + 10x^2 + C1)dx = (1/(4+1))4x^(4+1) + (1/(3+1))4x^(3+1) + (1/(2+1))10x^(2+1) + C1x + C2
Simplifying the integral, we get:
f(x) = (4/5)x^5 + x^4 + (10/3)x^3 + C1x + C2
Step 3: Apply the initial conditions
Now, we can use the given initial conditions f(0) = 9 and f(1) = 12 to determine the values of C1 and C2.
Given: f(0) = 9
Substituting x = 0 into the equation f(x), we get:
9 = (4/5)(0)^5 + (0)^4 + (10/3)(0)^3 + C1(0) + C2
Simplifying, we find:
C2 = 9
Given: f(1) = 12
Substituting x = 1 into the equation f(x), we get:
12 = (4/5)(1)^5 + (1)^4 + (10/3)(1)^3 + C1(1) + 9
Simplifying, we find:
C1 = 4 - 14/3 = 2/3
Finally, substituting the values C1 = 2/3 and C2 = 9 back into the equation for f(x), we have:
f(x) = (4/5)x^5 + x^4 + (10/3)x^3 + (2/3)x + 9
Therefore, the function f(x) is given by:
f(x) = (4/5)x^5 + x^4 + (10/3)x^3 + (2/3)x + 9