Problem 22 - Calculus

Question

For \(F(x)=\int_{x}^{2 x} \sin t d t,\) locate \(F(\pi+\Delta x)-F(\pi)\) on a sine graph. Where is \(F(\Delta x)-F(0)\) ?

Step-by-Step Solution

Answer

\(F(\pi+\Delta x)-F(\pi)\) is a vertical shift by 2; \(F(\Delta x)-F(0)\) is the area between \(\cos(\Delta x)\) and \(\cos(2\Delta x)\).

Step 1: Understanding the problem

We start with the integral function given as \(F(x)=\int_{x}^{2x} \sin t \, dt\). We need to evaluate \(F(\pi+\Delta x) - F(\pi)\) and \(F(\Delta x) - F(0)\). These expressions involve calculating the integral over different ranges.

Step 2: Calculate F(x)

The function \(F(x)=\int_{x}^{2x} \sin t \, dt\) defines the area under the sine curve from \(x\) to \(2x\). We can rewrite this using a property of integrals as \(F(x) = \int_{0}^{2x} \sin t \, dt - \int_{0}^{x} \sin t \, dt\).

Step 3: Use the Fundamental Theorem of Calculus

Using the Fundamental Theorem of Calculus, the result for \(\int \sin t \, dt\) is \(-\cos t\). Thus, \(F(x)\) becomes \([-\cos (2x) - (-\cos (x))] = \cos x - \cos(2x)\).

Step 4: Compute F(π+Δx) - F(π)

We substitute \(x = \pi + \Delta x\) and \(x = \pi\) into \(F(x) = \cos x - \cos(2x)\). We find:- \(F(\pi + \Delta x) = \cos(\pi + \Delta x) - \cos(2(\pi + \Delta x))\)- \(F(\pi) = \cos(\pi) - \cos(2\pi)\)Then calculate the difference:\[F(\pi + \Delta x) - F(\pi) = (\cos(\pi + \Delta x) - \cos(2\pi + 2\Delta x)) - (\cos(\pi) - \cos(2\pi))\].

Step 5: Simplify using trigonometric identities

Using the trigonometric identities \(\cos(\pi + \Delta x) = -\cos(\Delta x)\) and \(\cos(2\pi + 2\Delta x) = \cos(2\Delta x)\), we simplify:\[F(\pi + \Delta x) - F(\pi) = (-\cos(\Delta x) - \cos(2\Delta x)) - (-1 - 1) = -\cos(\Delta x) - \cos(2\Delta x) + 2\].

Step 6: Compute F(Δx) - F(0)

Substitute \(x = \Delta x\) and \(x = 0\):- \(F(\Delta x) = \cos(\Delta x) - \cos(2 \Delta x)\)- \(F(0) = 1 - 1 = 0\)Then calculate the difference:\[F(\Delta x) - F(0) = \cos(\Delta x) - \cos(2\Delta x)\].

Step 7: Locate on the sine graph

The expressions \(F(\pi + \Delta x) - F(\pi) = -\cos(\Delta x) - \cos(2\Delta x) + 2\) and \(F(\Delta x) - F(0) = \cos(\Delta x) - \cos(2\Delta x)\) give areas under sine curves shifted and scaled by these trig identities. These fluctuations represent vertical shifts on the sine curve based on \(\Delta x\).

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a critical link between differentiation and integration, two main operations in calculus. It consists of two parts. The first part guarantees that the integral of a function can be reversed by a derivative. The second part provides a method to compute the definite integral of a function. It states that if a function is continuous over an interval, its integral on that interval is determined by its antiderivative. In our problem, we used the Fundamental Theorem to find an antiderivative for

• The integral of \( \sin t \) is \(-\cos t \), a notable trigonometric function often encountered in calculus problems.
• This assists in transitioning from an integral expression to a more analysable form \( F(x) = \cos x - \cos(2x) \).

The ability to switch between a function's integrals and its derivatives allows for more profound insights into the function's behavior, such as analyzing its area under curves.

Integral Functions

Integral functions arise when dealing with expressions involving integrals. They describe the accumulation of quantities, such as area under a curve. In our problem, the function \( F(x)=\int_{x}^{2x} \sin t \, dt \) represents the area under the sine curve from \( x \) to \( 2x \):

• We broke down the integral using the additivity property of integrals, rewriting it as \( \int_{0}^{2x} \sin t \, dt - \int_{0}^{x} \sin t \, dt \).
• This decomposition clarifies the change in area as \( x \) varies.

Operating on integral functions often involves simplifying or transforming expressions, estimated with antiderivatives or derivatives, helping reveal changes in the dependent variable over defined bounds.

Trigonometric Identities

Trigonometric identities are mathematical expressions connecting the angles and lengths in trigonometry. They simplify complex expressions into more manageable forms. In our solution, they were essential to simplifying the results:

• We used identities like \( \cos(\pi + \Delta x) = -\cos(\Delta x) \) and \( \cos(2\pi + 2\Delta x) = \cos(2\Delta x) \) to resolve the formula \( F(\pi + \Delta x) - F(\pi) \).

These transformations:

• Reduce overall complexity.
• Allow easier calculation by converting to less abstract terms.

Trigonometric identities are invaluable tools in calculus. They are employed to make integration and differentiation of trigonometric functions more intuitive. This simplification was crucial for resolving the integral functions in the exercise.