/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 81 Let \(F(x)=\int_{a}^{u(x)} f(t) ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 81

Let \(F(x)=\int_{a}^{u(x)} f(t) d t\) for the specified \(a, u,\) and \(f .\) Use a CAS to perform the following steps and answer the questions posed. a. Find the domain of \(F .\) b. Calculate \(F^{\prime}(x)\) and determine its zeros. For what points in its domain is \(F\) increasing? Decreasing? c. Calculate \(F^{\prime \prime}(x)\) and determine its zero. Identify the local extrema and the points of inflection of \(F .\) d. Using the information from parts \((a)-(c),\) draw a rough hand-sketch of \(y=F(x)\) over its domain. Then graph \(F(x)\) on your CAS to support your sketch. $$ a=0, \quad u(x)=1-x, \quad f(x)=x^{2}-2 x-3 $$

Short Answer

Expert verified
F(x) is decreasing on \((-\infty, 2-\sqrt{2}) \cup (2+\sqrt{2}, 1)\); \(x=2\) is the inflection point.

Step by step solution

01

Define F(x)

Given the function \(F(x) = \int_{0}^{1-x} (t^2 - 2t - 3) \, dt\), this is an integral from constant \(a = 0\) to \(u(x) = 1 - x\). We need to first solve the integral for the variable limit to define \(F(x)\).
02

Evaluate the Integral

Calculate \(\int (t^2 - 2t - 3) \, dt\) to find the antiderivative: \( \frac{t^3}{3} - t^2 - 3t + C \). Evaluate from \(0\) to \(1-x\). This gives \(F(x) = \bigg[\frac{(1-x)^3}{3} - (1-x)^2 - 3(1-x)\bigg] - [0]\).
03

Simplify F(x)

Simplify \(F(x)\) to get it in a standard polynomial form: \(F(x) = \frac{(1-x)^3}{3} - (1-x)^2 - 3(1-x)\). When expanded, this becomes \(F(x) = \frac{-x^3}{3} + \frac{4x^2}{2} - \frac{9x}{3}\).
04

Determine the Domain of F(x)

Since \(u(x) = 1 - x\) must be non-negative, the domain of \(F\) is \(-\infty < x \leq 1\). This bounds the input values such that the upper limit of the integral remains feasible.
05

Compute F'(x) using FTC

Using the Fundamental Theorem of Calculus, \(F'(x) = f(u(x)) \cdot u'(x) = ((1-x)^2 - 2(1-x) - 3) \cdot (-1)\).
06

Simplify and Find Zeros of F'(x)

Simplify \(F'(x) = -(x^2 - 4x + 2)\). To find the zeros, solve the equation \(x^2 - 4x + 2 = 0\) using the quadratic formula, resulting in \(x = 2 \pm \sqrt{2}\).
07

Determine Increasing/Decreasing Intervals

The sign of \(F'(x)\) determines where \(F\) is increasing or decreasing. If \(F'(x) > 0\), \(F\) is decreasing (because of the negative sign), and if \(F'(x) < 0\), \(F\) is increasing. Analyze around the zeros to determine intervals.
08

Compute F''(x)

Differentiate \(F'(x)\) to get \(F''(x) = -2x + 4\). Simplifying gives \(F''(x) = -2(x - 2)\).
09

Determine Zero of F''(x)

Set \(F''(x) = 0\) and solve for \(x\), yielding \(x = 2\). This point indicates a potential inflection point.
10

Analyze Extrema and Inflection Points

Check sign changes in \(F''(x)\) around \(x = 2\) to see if it's a point of inflection. No changes indicate no local extrema or points of inflection within the domain.
11

Sketch F(x)

Using the domain, zeros, and signs from \(F'(x)\) and \(F''(x)\), sketch \(F(x)\) to emphasize increasing and decreasing parts. Validate with a CAS graph to ensure accuracy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus establishes a profound connection between differentiation and integration. It essentially states that if a function is continuous on a closed interval \[a, b\], then its integral over that interval is linked to its antiderivative. This theorem is split into two parts:
  • The first part tells us how to differentiate an integral function which, in our exercise, is expressed as \(F(x) = \int_{0}^{1-x} f(t) \, dt\).
  • The second part enables us to compute a definite integral if we know its antiderivative.
In practice, when tasked with finding \(F'(x)\) from such an expression, the theorem gives us that \(F'(x) = f(u(x)) \cdot u'(x)\), where \(u(x)\) is the variable upper limit of integration. This simplifies the process of finding the derivative of the integral function.
Polynomial Functions
A polynomial function is a type of mathematical expression consisting of variables raised to integer powers and different coefficients. In our exercise, the function under consideration is \(f(t) = t^2 - 2t - 3\). This represents a quadratic polynomial function.

A few characteristics of polynomial functions include:
  • The degree of the polynomial, which is the highest power of the variable in the polynomial. Here it is 2, indicating a quadratic function.
  • Polynomial functions are continuous and have smooth turns, making them straightforward to integrate and differentiate.
  • They are easy to simplify and expand, necessary for our calculation of \(F(x)\) in the exercise.
This polynomial's behavior over certain intervals can be analyzed using derivatives, as demonstrated in the later steps of analyzing \(F(x)\).
Critical Points
Critical points of a function are where its derivative is zero or undefined. These points are fundamental in determining the behavior of functions, specifically in identifying maxima, minima, and inflection points. In our exercise, \(F'(x)\) was found to determine critical points by setting the derivative equal to zero:

  • Solving \(x^2 - 4x + 2 = 0\) using the quadratic formula gives critical points at \(x = 2 \pm \sqrt{2}\).
Critical points help us analyze intervals where \(F(x)\) is increasing or decreasing by taking the derivative鈥檚 sign into account.
  • If \(F'(x) > 0\), the function is increasing.
  • If \(F'(x) < 0\), the function is decreasing.
Analyzing these critical points is essential in forming a sketch of the function and understanding its overall trend beyond just single points.
Graphical Analysis
Graphical analysis involves using visual means to understand a function's properties by examining its graph. For our function \(F(x)\), this involves plotting the curve to see how it behaves over its domain. The goal here is to:
  • Identify regions of increase or decrease using \(F'(x)\).
  • Find potential points of inflection using \(F''(x)\).
  • Validate these findings through sketching.
  • Use technology tools like a CAS (Computer Algebra System) for more precise graphs.
Through graphical analysis, the theoretical insights gained from formulas can be translated into visual understanding, thereby ensuring better comprehension of how the function behaves. A visual graph reveals trends, periodicity, and other behaviors that might not be immediately evident from equations alone.
In summary, this approach solidifies our analytical findings and helps us predict the behavior of a function across its domain.

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Most popular questions from this chapter

Use the Substitution Formula in Theorem 7 to evaluate the integrals. $$ \text { a. } \int_{0}^{1} \frac{x^{3}}{\sqrt{x^{4}+9}} d x \quad \text { b. } \int_{-1}^{0} \frac{x^{3}}{\sqrt{x^{4}+9}} d x $$

Find the areas of the regions enclosed by the lines and curves. $$ y=\left|x^{2}-4\right| \quad \text { and } \quad y=\left(x^{2} / 2\right)+4 $$

Integrals of nonpositive functions Show that if \(f\) is integrable then $$ f(x) \leq 0 \quad \text { on } \quad[a, b] \quad \Rightarrow \quad \int_{a}^{b} f(x) d x \leq 0 $$

Find the areas of the regions enclosed by the lines and curves. $$ y=x^{4}-4 x^{2}+4 \text { and } y=x^{2} $$

Find the areas of the regions enclosed by the lines and curves. $$ y=2 x-x^{2} \text { and } y=-3 $$
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