Suppose that f is continuously differentiable from x=a to...

Suppose that $f$ is continuously differentiable from $x=a$ to $x=b .$ Show that the
length of the graph of $f=\int_{a}^{b}|\sec [\alpha(x)]| d x$ where $\alpha(x)$ is the inclination of the tangent line at $(x, f(x))$

Key Concepts

Trigonometric Identity Relating Secant and Tangent

In trigonometry, the identity sec^2? = 1 + tan^2? links the secant and tangent functions. When applied to a differentiable function where the derivative is interpreted as the tangent of the angle of the tangent line, this identity facilitates rewriting the integral for the arc length. It shows that the conventional arc length formula agrees with the expression given via the secant of the inclination angle.

Arc Length of a Curve

The arc length of a curve defined by a function is a measure of the distance along the curve between two points. In calculus, if a function is continuously differentiable, its arc length from x = a to x = b is calculated using the integral of the square root of (1 plus the square of the derivative) over the interval. This concept is central to understanding how the geometry of a curve connects to its analytical expression.

Continuously Differentiable Functions

A continuously differentiable function is one that not only has a derivative at every point in its domain but also has a derivative that is continuous. This property guarantees that the function behaves smoothly without any abrupt changes, which is essential when applying integral calculus to find attributes like the arc length of its graph.

Inclination of the Tangent Line

The inclination of the tangent line at a given point on a curve is the angle that the tangent makes with the positive x-axis. This angle provides a geometric interpretation of the derivative because the tangent’s slope is the tangent of the angle of inclination. Understanding this relationship is key to connecting the derivative of the function with trigonometric functions such as the secant.

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