Question

State theorems analogous to Theorems 3.3 .10 and 3.3 .11 for the function $$ G(x)=\int_{x}^{c} f(t) d t $$ and show how your theorems can be obtained from them.

Short answer

Question: State the analogous versions of Theorems 3.3.10 and 3.3.11 for the function $$ G(x)=\int_{x}^{c} f(t) dt $$ and briefly explain how to derive them from their original counterparts. Answer: For the function $$ G(x)=\int_{x}^{c} f(t) dt $$, the analogous versions of Theorems 3.3.10 and 3.3.11 are: 1. If f(x) is continuous on an interval [a, c], then G'(x) = -f(x). 2. If f(x) is differentiable on an interval [a, c] and its derivative f'(x) is continuous on [a, c], then G''(x) = -f'(x). To derive these theorems from their original counterparts: 1. For the first analogue theorem, define the function $$ H(x) =\int_{a}^{x} f(t) dt $$, which has the property $$ H'(x)=f(x) $$ due to Theorem 3.3.10. Express G(x) in terms of H(x) as $$ G(x) = -H(x) $$, and differentiate it with respect to x to obtain $$ G'(x) = -f(x) $$. 2. For the second analogue theorem, use the same definition for H(x) and the property $$ H''(x)=f'(x) $$ due to Theorem 3.3.11. Differentiate the expression $$ G(x) = -H(x) $$ twice with respect to x to obtain $$ G''(x) = -f'(x) $$.

Step-by-step solution

Theorem for G(x) similar to Theorem 3.3.10

Since G(x) is given as $$ G(x) =\int_{x}^{c} f(t) dt $$, the first analogous theorem should define the derivative G'(x) in terms of f(x). To do that, we need to use Leibniz's rule for differentiation under the integral sign: $$ G'(x) = -f(x) $$

Theorem for G(x) similar to Theorem 3.3.11

Similarly, for the second analogous theorem, we will state the expression for the second derivative G''(x) in terms of f'(x). Based on the result of the previous theorem, we have: $$ G''(x) = -f'(x) $$ Now we will demonstrate how to obtain these theorems from their original counterparts, Theorems 3.3.10 and 3.3.11.

Deriving the First Analogue Theorem

Assume that f(x) is continuous on an interval [a, c]. Let $$ H(x) =\int_{a}^{x} f(t) dt $$ which has the property of $$ H'(x)=f(x) $$ due to Theorem 3.3.10. Now we can write $$ G(x) = \int_{x}^{c}f(t) dt = - \int _{c}^x f(t) dt.$$ If we rewrite the integration limits of H(x) by swapping a and c, we get $$ -H(x) = \int_{c}^{x} f(t) dt = G(x). $$ Differentiating both sides regarding x, we obtain: $$ -H'(x)=G'(x) $$ Since H'(x) = f(x), we have: $$ -f(x)=G'(x) $$ which is the analogous theorem to 3.3.10 for G(x).

Deriving the Second Analogue Theorem

Once again, assume that f(x) is differentiable on an interval [a, c] and its derivative f'(x) is continuous on the same interval. Let $$ H(x) =\int_{a}^{x} f(t) dt $$ which has the property of $$ H''(x)=f'(x) $$ due to Theorem 3.3.11. We have shown that $$ -H(x) = G(x) $$ and therefore, differentiating the expression regarding x twice, we get: $$ -H''(x)=G''(x) $$ Since H''(x) = f'(x), we have: $$ -f'(x)=G''(x) $$ which is the analogous theorem to 3.3.11 for G(x).

Key concepts

Leibniz's Rule

Leibniz's Rule is fascinating because it elegantly bridges the gap between differentiation and integration, two fundamental operations in calculus. Named after the renowned mathematician Gottfried Wilhelm Leibniz, this rule provides a formula for differentiating an integral whose limits are functions of the differentiation variable - a concept that might sound perplexing at first, but let's break it down. Imagine a moving slice of a cake: the position of the slice changes as you adjust the size of the piece you’re cutting. In mathematical terms, when the boundaries of an integral change with the variable of differentiation, Leibniz's Rule helps us understand how that 'slice' evolves as well.

Here’s the gist of it: if you have a function integrated between two bounds, say from function u(x) to v(x), and these bounds themselves change as x changes, how then does the integral's value change concerning x? Leibniz's Rule gives us a precise answer and it becomes especially crucial when handling complex functions in real analysis. It dictates that the derivative of such an integral is the difference between the function evaluated at the upper bound and multiplied by the derivative of the upper bound, and the function evaluated at the lower bound multiplied by the derivative of the lower bound, plus the integral of the partial derivative of the function with respect to x over the interval from the lower to the upper bound. In formula form, for a continuous function f(t,x), it looks like this:
\[\begin{equation} \frac{d}{dx} \bigg(\int_{u(x)}^{v(x)} f(t,x) dt\bigg) = f(v(x),x) \times v'(x) - f(u(x),x) \times u'(x) + \int_{u(x)}^{v(x)} \frac{\partial f(t,x)}{\partial x} dt. \end{equation}\]
By using Leibniz's Rule, complex problems involving variable limits of integration can be solved, enhancing our understanding of how functions behave when bounded by other functions.

Differentiation Under the Integral Sign

When we talk about 'differentiation under the integral sign', we're venturing into a technique that unlocks a new realm of problem-solving skills in calculus. This fancy term essentially means we take the derivative of an integral when the integrated function involves a variable upon which we are differentiating. It's akin to having a machine that wraps gifts, and you're changing the pattern of the wrapping paper while it's operating - the output, that is, the wrapped gift, starts showing the new pattern as it emerges.

To comprehend this concept, picture an integral that expresses the accumulation of a quantity, like the total distance traveled over time. Now imagine you wish to find out how this total accumulation rate changes at an instant - in other words, you're after the instantaneous speed. Differentiation under the integral sign allows you to determine this rate, effectively 'peeling back' the layers of accumulation to expose the rate at a particular instant. It's a tool that proves extremely useful in various fields of mathematics and physics because it simplifies what could otherwise be a very complex problem.

The exercise provided showcases how this concept can turn a potentially tough integration problem into a more manageable one. Given the function G(x) which is an integral with a variable lower limit, we apply differentiation under the integral sign to find the derivatives of G(x). By reversing the limits and tweaking the sign, we use the known theorems for functions integrated from a constant to x, and then directly apply differentiation under the integral to obtain the desired results for G'(x) and G''(x).

Continuous Functions

In the realm of real analysis, continuous functions play the role of a smooth sea voyage: there's no jumping from point to point; it's a steady sail with no abrupt changes in direction or interruptions. A function is continuous if, at every point within its domain, the value of the function approaches the function's actual value at that point, as you zoom in infinitely close. In simpler terms, you can draw it without lifting your pen off the paper.

But why do continuous functions matter so much? They form the bedrock upon which the edifice of calculus stands. The continuity of a function ensures the reliability of limits, derivatives, and integrals, which are the core tools of calculus. Consider a real-world scenario, like predicting the trajectory of a ball thrown into the air. If we model this situation with a function that measures the height of the ball over time, a continuous function would mean no unexpected teleportation of the ball from one point to another - it follows a path dictated by the forces of physics, smoothly transitioning through each moment.

In our exercise, we assume the continuity of the function f(x) over a specific interval. This assumption is essential because it allows us to use the fundamental theorems of calculus, which guarantee that we can differentiate and integrate f(x) in a meaningful way. In practice, this continuity means that as we explore the behavior of G(x), which is an integral with a variable bound, we can seamlessly apply differentiation - thanks to Leibniz’s rule and differentiation under the integral sign - without worrying about any 'jumps' or 'holes' in the function's graph that could make the problem unsolvable.