Find f if f′(x)=4/sqrt(1−x^2) and f(12)=5.
since d/dx arcsin(x) = 1/√(1-x^2) you get
4arcsin(x) + C
Unfortunately, arcsin(12) is undefined. Maybe you are one of those people who cannot figure out how to type fractions.
4arcsin(1/2) + C = 5
4 * π/6 + C = 5
now finish it off
To find the function f, we need to integrate f′(x) and solve for the constant of integration using the given initial conditions.
Given: f′(x) = 4/√(1−x^2)
To integrate f′(x), we will use the substitution method. Let's substitute u = 1-x^2:
Differentiating both sides with respect to x:
du/dx = -2x
Solving for dx, we have:
dx = du / (-2x)
Now, let's substitute these values into the integral:
∫ (4/√(1−x^2)) dx = ∫ (4/√u) (du / (-2x))
Simplifying the integral:
(-1/2) ∫ (4/√u) du
Taking the constant factors outside the integral:
(-2) ∫ (1 / √u) du
Now, let's integrate the right side of the equation:
(-2) ∫ (u^(-1/2) du)
= (-2) (2u^(1/2)) + C_1
= -4√u + C_1
Substituting back the value of u = 1-x^2:
= -4√(1-x^2) + C_1
Finally, to find the particular solution f(x), we need to determine the value of the constant C_1 using the given condition f(12) = 5.
Given: f(12) = 5
Substituting x = 12 into the derived expression of f(x):
-4√(1-12^2) + C_1 = 5
Simplifying the square root expression:
-4√(1-144) + C_1 = 5
-4√(-143) + C_1 = 5
Since the expression inside the square root is negative, there is no real number solution for this equation. Therefore, there is no function f(x) that satisfies f′(x) = 4/√(1−x^2) and f(12) = 5.
To find the function f(x), we need to integrate the derivative f′(x). In this case, the derivative is given as f′(x) = 4/sqrt(1 − x^2).
Step 1: Start by integrating the given derivative f′(x):
∫(4/sqrt(1 − x^2)) dx = ?
Step 2: To simplify the integration, we can rewrite the denominator as (1 − x^2)^(1/2). In order to integrate this expression, we can use a trigonometric substitution.
Let x = sin(θ).
Then, dx = cos(θ) dθ, and 1 − x^2 = 1 − sin^2(θ) = cos^2(θ).
Substituting these values into the integral:
∫(4/(cos(θ))) * (cos(θ) dθ)
Step 3: Simplify the integral:
∫4 dθ = 4θ + C
Step 4: Convert θ back to x:
θ = arcsin(x)
Step 5: Substitute the values:
f(x) = 4 * arcsin(x) + C
Step 6: Use the given condition f(12) = 5:
5 = 4 * arcsin(12) + C
Step 7: Solve for C:
C = 5 - 4 * arcsin(12)
Step 8: Final answer:
The function f(x) is given by f(x) = 4 * arcsin(x) + (5 - 4 * arcsin(12)).