Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 1 - Functions

Problem 104

Chapter 1, Problem 104

{Use of Tech} Triple intersection Graph the functions f(x) = x³,g(x)=3^x, and h(x)=x^x and find their common intersection point (exactly).

Verified step by step guidance

Step 1: Understand the problem by identifying the functions involved: f(x) = x^3, g(x) = 3^x, and h(x) = x^x. We need to find the common intersection point of these three functions.

Step 2: Set up the equations for intersection by equating the functions pairwise: f(x) = g(x), g(x) = h(x), and f(x) = h(x). This will help us find the x-values where the functions intersect.

Step 3: Solve the equation f(x) = g(x), which is x^3 = 3^x. This involves finding the x-value(s) where the cubic function equals the exponential function.

Step 4: Solve the equation g(x) = h(x), which is 3^x = x^x. This involves finding the x-value(s) where the exponential function equals the power function.

Step 5: Solve the equation f(x) = h(x), which is x^3 = x^x. This involves finding the x-value(s) where the cubic function equals the power function. The common solution to all three equations will be the intersection point.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Functions

Graphing functions involves plotting their values on a coordinate system to visualize their behavior. For the functions f(x) = x³, g(x) = 3^x, and h(x) = x^x, understanding their shapes and intersections is crucial. Each function has distinct characteristics: f(x) is a polynomial, g(x) is an exponential function, and h(x) is a power function, which influences how they intersect.

Intersection Points

Intersection points of functions occur where their outputs are equal, meaning f(x) = g(x) = h(x). To find these points, one typically sets the equations equal to each other and solves for x. This process may involve algebraic manipulation or numerical methods, especially when dealing with complex functions like x^x.

Numerical Methods

Numerical methods are techniques used to approximate solutions to equations that cannot be solved analytically. In the context of finding intersection points, methods such as the Newton-Raphson method or bisection method can be employed to find roots of the equations. These methods are particularly useful when dealing with transcendental functions like 3^x and x^x, where exact solutions may be difficult to obtain.

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Textbook Question

Design a sine function with the given properties.

It has a period of $12$ with a minimum value of $-4$ at $t=0$ and a maximum value of $4$ at $t=6$ .

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Textbook Question

Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle $\theta$ (measured in radians) is $A=$\frac$12r^2$\theta$$ .

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Textbook Question

Identify the amplitude and period of the following functions.

$q$\left$(x$\right$)=3.6$\cos\[\left$($\frac{\pi x}{24}\]\right$)$

Textbook Question

Beginning with the graphs of $y=$\sin$ x$ or $y=$\cos$ x$ , use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.

$q$\left$(x$\right$)=3.6$\cos\[\left$($\frac{\pi x}{24}\]\right$)+2$

Textbook Question

Daylight function for 40 °N Verify that the function $D(t)=2.8$\sin$($\frac{2\pi}{365}$(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset. It has a period of 365 days.

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