Question

Functions defined by integrals Consider the function $g$, which is given in terms of a definite integral with a variable upper limit. a. Graph the integrand. b. Calculate $g^{\prime }(x)$ c. Graph g, showing all your work and reasoning. $$g(x)={\int }_0^x\sin \left(\pi t^2\right)dt\text{ (a Fresnelintegral) }$$

Short answer

Question: Sketch the graph of $g(x)={\int }_0^x\sin \left(\pi t^2\right)dt$ and find its derivative $g^{\prime }(x)$. Answer: The graph of the function $g(x)$ is an oscillating curve, resembling a sine function, but with an envelope influenced by the squared term in the integrand. The derivative of $g(x)$ is: $g^{\prime }(x)=\sin \left(\pi x^2\right)$.

Step-by-step solution

Graph the integrand function

To graph the integrand function, which is $\sin \left(\pi t^2\right)$, we must understand its behavior between the integration limits. In our case, we have the integration limits from $0$ to $x$. By observing the function, we can see that it has periodic oscillations due to the sine function but with varying frequency due to the square term inside. You can either sketch the graph by hand, noting the important points and behavior, or you can use graphing software like Desmos or GeoGebra to generate a graph for this function. Once you have the graph, you'll notice the sine wave pattern with increasing frequency as the value of $t$ increases.

Calculate the derivative of $g(x)$, which is $g^{\prime }(x)$

To find the derivative of $g(x)$, we'll use the Fundamental Theorem of Calculus part 1, which states: $$\frac{d}{dx}{\int }_{a(x)}^{b(x)}f(t)dt=f(b(x))b^{\prime }(x)-f(a(x))a^{\prime }(x)$$ In our case, we have: - $a(x)=0$: the lower limit which is a constant - $b(x)=x$: the variable upper limit - $f(t)=\sin \left(\pi t^2\right)$: the integrand function Applying the theorem, we have: $$g^{\prime }(x)=\sin \left(\pi x^2\right)\cdot 1-\sin (\pi (0{)}^2)\cdot 0=\sin \left(\pi x^2\right)$$ So, the derivative of $g(x)$ is: $$g^{\prime }(x)=\sin \left(\pi x^2\right)$$

Graph the function $g(x)$, showing all the work and reasoning

Graphing $g(x)$ can be quite tricky, as it involves plotting the integral of a function. Since we don't have a closed-form expression for $g(x)$, we have to approximate its graph. In this case, g(x) can be referred to as a Fresnel integral function. One way to graph $g(x)$ is by computing the integral numerically (using some numerical integration technique like Riemann sums, trapezoidal rule, or Simpson's rule) for several values of $x$. Then, plot the resulting points on a graph. Another way to graph $g(x)$ is by using graphing software or tools like Desmos, GeoGebra, or WolframAlpha. You can enter the function $g(x)={\int }_0^x\sin \left(\pi t^2\right)dt$, and the graphing tool will return an approximate graph for the function. Regardless of the method you use, the graph of $g(x)$ will show a curve that oscillates similarly to the sine function but with an envelope that is influenced by the squared term in the integrand.

Key concepts

Fresnel integral

The Fresnel integral is a unique mathematical expression involving integration. It doesn’t behave like regular functions and is often used in optics and wave physics. In particular, it defines a curve based on an integral of sine or cosine functions, with a quadratic term included in the sine or cosine arguments. Fresnel integrals usually appear in terms like $C(x)={\int }_0^x\cos (\pi t^2)dt$ and $S(x)={\int }_0^x\sin (\pi t^2)dt$.

These integrals are used in path calculations in optics, specifically in analyzing the way light waves and shadows interact. They create beautiful, complex curves that don’t have straightforward expressions. Instead, their graphs are approximations that require numerical methods to visualize. For students delving into calculus, the Fresnel integral is an example of how integrals can define new, interesting functions that go beyond typical linear or polynomial shapes.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, two core concepts of calculus. It backbones the process of evaluating definite integrals and lends a tool to differentiate functions defined by integrals.

According to the first part of this theorem, if you have an integral defined like $g(x)={\int }_a^{x}f(t)dt$, its derivative $g^{\prime }(x)$ is simply the function $f(x)$. This is exceptionally powerful because it provides a pathway to calculate the derivative without evaluating the integral directly.

In our specific problem, where $g(x)={\int }_0^x\sin (\pi t^2)dt$, the theorem pushes us directly to $g^{\prime }(x)=\sin (\pi x^2)$. Thus, no matter how complex the integrand looks, differentiation returns us to the immediate function being integrated, provided the limits match the conditions needed.

Numerical Integration

Numerical integration is the process of approximating the value of an integral where the exact outcome is complex, difficult, or impossible to find using basic calculus. This is particularly useful when dealing with functions or integrals that have no closed-form antiderivative, like our Fresnel integral scenario.

The concept involves various techniques to estimate the integral:

• Riemann Sums: Splitting the area under a curve into simple shapes (rectangles) and summing their areas.
• Trapezoidal Rule: Approximating the curve as a series of trapezoids rather than rectangles.
• Simpson’s Rule: Using parabolic segments instead of straight lines to approximate the area.

These methods are fundamental in computational mathematics, allowing computers to generate approximations of integrals when no algebraic solution exists. In practical scenarios, one can use software to perform these calculations and plot the corresponding graph for deeper analysis.

Graphing Functions

Graphing functions provides a visual understanding of mathematical behaviors, aiding in interpreting the effects of variables on equations. When dealing with integrals, especially those defined as functions with variable upper limits, graphing is a suitable method to comprehend complex behaviors.

For functions like our Fresnel-type integral $g(x)={\int }_0^x\sin (\pi t^2)dt$, graphing becomes crucial. Since there’s no simple expression to describe $g(x)$, plotting helps us visualize how $g(x)$ behaves over different domains.

Techniques to graph such functions include:

• Using graphing calculators or software such as Desmos, GeoGebra, or WolframAlpha to input the integral and observe graphs dynamically.
• Calculating numerical integration values at various $x$ points and plotting them to create a rough sketch.

Graphing brings clarity, showing oscillations or patterns intrinsic to the integrand’s nature, and is an indispensable tool for understanding calculus at a deeper, visual level.