Final answer:
To find the length of the graph of f(x) = ln(5sec(x)) for 0 ≤ x ≤ π/3, we need to evaluate the integral of the absolute value of the derivative of f(x) over the given interval.
Step-by-step explanation:
To find the length of the graph of f(x) = ln(5sec(x)) for 0 ≤ x ≤ π/3, we need to evaluate the integral of the absolute value of the derivative of f(x) over the given interval.
Step 1: Find the derivative of f(x) using the chain rule. The derivative of ln(u) is 1/u * u', so the derivative of ln(5sec(x)) is 1/(5sec(x)) * (5sec(x)tan(x)).
Step 2: Take the absolute value of the derivative of f(x) to find the length of the graph. The absolute value of 1/(5sec(x)) * (5sec(x)tan(x)) is |tan(x)|.
Step 3: Evaluate the integral of |tan(x)| over the interval 0 to π/3. This can be done using integration techniques or using a graphing calculator. The result will give the length of the graph of f(x).
The student is asking to find the length of the graph of the function f(x) = ln(5sec(x)) over the interval from 0 to π/3. This requires calculating the arc length of the curve represented by the function. The formula for the arc length of a function f(x) from a to b is given by:
L = ∫_a^b √(1 + [f'(x)]^2) dx
In this case, we first need to find the derivative of f(x), and then integrate the square root of 1 plus the square of that derivative over the interval from 0 to π/3. It involves the application of calculus concepts like differentiation and integration.