Question

Evaluate the following definite integrals. If \(f(x)=\int_{-\pi / 4}^{x} \tan ^{2}(t) d t,\) find \(f^{\prime}\left(\frac{\pi}{6}\right)\).

Short answer

Answer: \(f'\left(\frac{\pi}{6}\right) = \frac{1}{3}\).

Step-by-step solution

Identify the function to differentiate

We are given the function \(f(x) = \int_{-\pi / 4}^{x} \tan ^{2}(t) d t\) and asked to find its derivative \(f'(x)\).

Differentiate using the Fundamental Theorem of Calculus

According to the fundamental theorem of calculus, the derivative of an integral of a function with respect to its upper limit is the function itself. So, we have $$f'(x) = \frac{d}{dx} \int_{-\pi / 4}^{x} \tan^2(t) dt = \tan^2(x).$$

Evaluate the function at the given point

Now, we need to find the value of \(f'(x)\) at \(x = \frac{\pi}{6}\). Using the formula from Step 2, we have $$f'\left(\frac{\pi}{6}\right) = \tan^2\left(\frac{\pi}{6}\right).$$ Since \(\tan\left(\frac{\pi}{6}\right) = \sqrt{3}/3\), we substitute this value into the above equation:

Final answer

The final answer is given by $$f'\left(\frac{\pi}{6}\right) = \left(\frac{\sqrt{3}}{3}\right)^2 = \boxed{\frac{1}{3}}.$$

Key concepts

Definite Integrals

A definite integral represents the area under the curve of a function on a specific interval. It's expressed as \(\int_{a}^{b}f(x)dx\), with \(a\) and \(b\) being the lower and upper limits of integration, respectively. In the context of the given exercise, the definite integral \(\int_{-\pi / 4}^{x} \tan ^{2}(t) dt\) calculates the area under the curve of \(\tan^2(t)\) from \( -\pi / 4\) to \(x\).

To better understand how to work with definite integrals, AP Calculus AB students are encouraged to visualize them as the accumulation of quantities. This can make applications in physics, like calculating displacements over a time interval, more comprehensible. Definite integrals also play a crucial role in other disciplines such as economics, engineering, and biology, where they are used to model and analyze various phenomena.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to an independent variable. The derivative is the slope of the tangent line to the function's graph at any given point. In our exercise, we are asked to find \(f'(x)\), the derivative of the function \(f(x)\), which is itself a definite integral.

The Fundamental Theorem of Calculus connects differentiation and integration by stating that the derivative of an integral with respect to its upper limit is the integrand evaluated at that upper limit. Thus, for the function \(f(x) = \int_{-\pi / 4}^{x} \tan ^{2}(t) dt\), the derivative \(f'(x) = \tan^2(x)\), indicating how the area under the \(\tan^2(t)\) curve changes as \(x\) changes. This theorem is a powerful tool because it simplifies the calculation of derivatives and helps students understand how these two fundamental calculus concepts are inversely related.

AP Calculus AB

AP Calculus AB is an advanced placement course designed for high school students with an in-depth focus on the concepts of limits, derivatives, definite integrals, and the Fundamental Theorem of Calculus. It serves as an introduction to college-level calculus and often provides students with college credits upon successful completion of the AP exam.

The problem involving \(f(x) = \int_{-\pi / 4}^{x} \tan ^{2}(t) dt\) is a typical example of the level of understanding required for the AP Calculus AB curriculum. Proficiency in working with definite integrals, differentiation, and applying the Fundamental Theorem of Calculus is essential for doing well in the course. Students should be able to not only calculate derivatives and integrals but also interpret their meanings and apply them to real-world scenarios, such as in physics and engineering problems.