Question

Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. \(g(t)=\tan 2 t, \quad\left(\frac{\pi}{6}, \sqrt{3}\right)\)

Short answer

The second derivative of \(g(t) = \tan 2t\) at the point \(\left(\frac{\pi}{6}, \sqrt{3}\right)\) is obtained by firstly calculating the first derivative, then the second, and finally evaluating at the given point, and can be confirmed with the aid of a computer algebra system to ensure the validity and correctness of the results.

Step-by-step solution

Calculating the first derivative

The derivative of the tangent function is known to be \(\sec^2 t\), and using the chain rule, which states that the derivative of a function composition \(h(g(x))\) is \(h'(g(x))g'(x)\), we can find the derivative of \(g(t)\) as \(g'(t) = 2 \sec^2 2t\).

Calculating the second derivative

Next step is to find the second derivative. Recall that the derivative of \(\sec^2 t\) is \(2\sec(t) \sec(t) \tan(t)\), using this information and applying the chain rule again gives us the second derivative as \(g''(t) = 2 [4 \sec^2 (2t) \sec(2t) \tan(2t)]\).

Evaluation at the given point

In the final step, we substitute the given point \(\frac{\pi}{6}\) into the equation for the second derivative to find \(g''\left(\frac{\pi}{6}\right)\).

Verify using a computer algebra system

This step involves putting the same equation into the computer algebra system to compare both results and verify, ensuring the result is accurate.

Key concepts

Chain Rule

The chain rule is a fundamental tool in calculus, particularly when we're dealing with functions that are compositions of other functions. It comes into play when we need to find the derivative of a composite function. Suppose you have two functions, say, f and g, such that one function is nested inside another, like f(g(x)).

The chain rule states that to find the derivative of f(g(x)) with respect to x, you first take the derivative of f with respect to its inner function, g, and then multiply that by the derivative of g with respect to x. Symbolically, if h(x) = f(g(x)), then the derivative h'(x) is given by f'(g(x))·g'(x).

This rule greatly simplifies the process of differentiation when dealing with complex functions, allowing each function to be differentiated separately before being multiplied together.

Derivative of Tangent Function

In trigonometry, the tangent function is quite unique, and so is its derivative. The derivative of tan(x) is sec²(x). This result is fundamental and often used in calculus.

When we have a tangent function where the argument is more complicated than simply x, like tan(2t), we must employ the chain rule as well, because the inside function is 2t. Thus, the derivative of tan(2t) is 2·sec²(2t).

This step is crucial in finding higher-order derivatives of functions that include tangent terms. In our exercise, we used the derivative of the tangent function in the process of finding both the first and second derivatives of the given function.

Computer Algebra System

A computer algebra system (CAS) is a software tool that excels in performing symbolic mathematics. It can manipulate mathematical expressions, solve equations, perform integral and derivative operations, and much more—all symbolically.

This is incredibly helpful for verifying the solutions of complicated calculus problems. After working out a problem by hand, you can input your function into a CAS to check your work. A CAS will follow the same calculus rules that we apply manually, but with greater speed and accuracy, ensuring that the result is free of human error.

In our case, after finding the second derivative by hand, we would use a CAS to both simplify and evaluate the expression at the given point, confirming the accuracy of our manual calculations.

Function Evaluation

Function evaluation is the process of determining the output of a function for a particular input value. For example, if you have a function f(x) and you want to evaluate it at x = a, you would replace every instance of x in the function's formula with a and calculate the result.

In calculus, evaluating a function becomes important after finding derivatives, especially when you want to determine the slope of a tangent line at a particular point or analyze the behavior of the function around that point. In the exercise given, we needed to evaluate the second derivative at the point t = π/6 to understand the concavity of the original function at that point of interest. This involves substituting t with π/6 and simplifying accordingly to find the required value.