(1 point) Find the particular antiderivative that satisfies the following conditions: p'(x) = -20/x^2 p(6) = 4
integral of dx/x^2 = integral x^-2 dx = -x^-1 + c = -1/x + c
so -20 [ -1/x + c ] = 20/x + some constant b
20/6 + b = 4
10/3 + b = 4
b = 12/3 - 10/3 = 2/3
so 20/x + 2/3
check my math !!!
To find the particular antiderivative (or primitive function) that satisfies the given conditions, we need to integrate the given derivative function and then apply the initial condition.
Step 1: Integration of p'(x)
To find p(x), we integrate p'(x) = -20/x^2 with respect to x. The indefinite integral of -20/x^2 is:
∫ -20/x^2 dx = -20 ∫ 1/x^2 dx
To integrate the term 1/x^2, we can use the power rule for integration: ∫ x^n dx = x^(n+1) / (n+1), where n ≠ -1.
For 1/x^2, we have n = -2, so:
∫ 1/x^2 dx = x^(-2+1) / (-2+1) = x^(-1) / (-1) = -1/x
Therefore,
∫ -20/x^2 dx = -20 ∫ 1/x^2 dx = -20 * (-1/x) = 20/x
So, p(x) = 20/x
Step 2: Applying the initial condition
To apply the initial condition p(6) = 4, we substitute x = 6 into the expression we found for p(x):
p(6) = 20/6 = 10/3
However, the initial condition states that p(6) = 4. To resolve this contradiction, we need to introduce a constant of integration, C.
p(x) = 20/x + C
Now, we can use the initial condition to find the value of C:
p(6) = 20/6 + C = 10/3 + C = 4
To isolate C, we subtract 10/3 from both sides:
C = 4 - 10/3
To simplify the right side, we need to find a common denominator:
C = 12/3 - 10/3 = 2/3
Therefore, the particular antiderivative (primitive function) that satisfies the conditions p'(x) = -20/x^2 and p(6) = 4 is:
p(x) = 20/x + 2/3
To find the particular antiderivative, we need to integrate the given function. Integrating with respect to x, we have:
∫(-20/x^2) dx
To integrate this function, we use the power rule for integration:
∫(x^n) dx = (x^(n+1))/(n+1) + C
Applying the power rule, we have:
∫(-20/x^2) dx = -20 ∫(1/x^2) dx
Using the power rule again, we have:
= -20 * (-1/x) + C
= 20/x + C
Now, we need to determine the particular antiderivative that satisfies the condition p(6) = 4. Plugging in x = 6, we can solve for the constant C:
20/6 + C = 4
Simplifying, we have:
10/3 + C = 4
Subtracting 10/3 from both sides:
C = 4 - 10/3
To simplify the right side, we get:
C = 12/3 - 10/3
C = 2/3
Therefore, the particular antiderivative that satisfies the given conditions is:
p(x) = 20/x + 2/3.