Finding the Function f(x) Given f'(x) and f(4) = 3

In this article, we will explore the process of finding the function f(x) when given the derivative f'(x) and a specific value of f(x). This type of problem is commonly encountered in high school and college-level calculus courses.

Given Information

• The derivative of the function f(x) is given as: f'(x) = cos(x) / x
• The value of the function f(x) at x = 4 is given as: f(4) = 3

Step 1: Understand the Problem

To find the function f(x), we need to use the given information about the derivative f'(x) and the value of f(4) to integrate and determine the expression for f(x).

Step 2: Integrate the Derivative

The derivative of a function f(x) is given as f'(x) = cos(x) / x. To find the function f(x), we need to integrate this expression.

Integrating both sides with respect to x, we get:

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

Using integration by parts, we can solve this integral:

Let u = cos(x), dv = 1/x dx Then, du = -sin(x) dx, v = ln(x)

Applying the integration by parts formula:

f(x) = [cos(x) ln(x)] - ∫ (-sin(x) ln(x)) dx f(x) = cos(x) ln(x) + ∫ sin(x) ln(x) dx

Now, we can use the substitution method to evaluate the remaining integral:

Let t = x, dt = dx Then, f(x) = cos(x) ln(x) + ∫ sin(t) ln(t) dt

Integrating the remaining integral, we get:

f(x) = cos(x) ln(x) - sin(x) ln(x) + C

Step 3: Determine the Constant of Integration

To find the constant of integration C, we can use the given value of f(4) = 3.

Substituting x = 4 into the expression for f(x), we get:

f(4) = cos(4) ln(4) - sin(4) ln(4) + C 3 = cos(4) ln(4) - sin(4) ln(4) + C C = 3 - cos(4) ln(4) + sin(4) ln(4)

Step 4: Write the Final Expression for f(x)

Substituting the value of the constant of integration C into the expression for f(x), we get the final function:

f(x) = cos(x) ln(x) - sin(x) ln(x) + 3 - cos(4) ln(4) + sin(4) ln(4)

This is the function f(x) that satisfies the given conditions of f'(x) = cos(x) / x and f(4) = 3.