교과서 질문
Functions and Graphs
Find the domain of y = (x + 3) / (4 − √(x² − 9)).
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
1장, 문제 1.1.12
Finding Formulas for Functions
A point P in the first quadrant lies on the graph of the function f(x) = √x. Express the coordinates of P as functions of the slope of the line joining P to the origin.
검증된 단계별 안내1
Identify the coordinates of point P on the graph of f(x) = √x. Since P lies on this graph, its coordinates can be expressed as (a, √a) for some positive value a.
Determine the slope of the line joining the origin (0, 0) to the point P (a, √a). The slope m is given by the formula m = (√a - 0) / (a - 0) = √a / a.
Express a in terms of the slope m. From the equation m = √a / a, we can rearrange to find a in terms of m: m = 1/√a, which implies √a = 1/m.
Solve for a by squaring both sides of the equation √a = 1/m to get a = 1/m².
Substitute a = 1/m² back into the coordinates of P. The x-coordinate is a = 1/m², and the y-coordinate is √a = 1/m. Therefore, the coordinates of P as functions of the slope m are (1/m², 1/m).
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Slope of a Line
The slope of a line measures its steepness and is calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points. For a line joining the origin (0,0) to a point P(x, f(x)), the slope can be expressed as f(x)/x. Understanding this concept is crucial for relating the coordinates of point P to the slope.
추천 영상:
05:13
Slopes of Tangent Lines
Function Representation
A function represents a relationship between inputs and outputs, typically expressed as f(x). In this case, f(x) = √x defines the output for any given input x. Recognizing how to manipulate and express functions is essential for deriving the coordinates of point P in terms of the slope.
추천 영상:
06:21Properties of Functions
Coordinate Geometry
Coordinate geometry involves the study of geometric figures using a coordinate system. In this context, the coordinates of point P are expressed as (x, √x). Understanding how to work with coordinates in the Cartesian plane is vital for solving problems related to slopes and functions.
추천 영상:
06:35Changing Geometries
관련 실천
교과서 질문
Radians and Degrees
On a circle of radius 10 m, how long is an arc that subtends a central angle of (a) 4π/5 radians? (b) 110°?
259
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교과서 질문
A hot-air balloon rising straight up from a level field is tracked by a range finder located 500 ft from the point of liftoff. Express the balloon’s height as a function of the angle the line from the range finder to the balloon makes with the ground.
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교과서 질문
Finding a Viewing Window
In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.
f(x) = (x² − 1)/(x² + 1)
270
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교과서 질문
Piecewise-Defined Functions
In Exercises 35 and 36, find the (a) domain and (b) range.
𝔂 = { √ -x, -4 ≤ x ≤ 0
{ √ x, 0 < x ≤ 4
249
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교과서 질문
Suppose that ƒ and g are both odd functions defined on the entire real line. Which of the following (where defined) are even? odd?
a. ƒg
b. ƒ³
c. ƒ(sin x)
d. g(sec x)
e. |g|
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