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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 1 - FunctionsProblema 1.1.10
Capitolo 1, Problema 1.1.10

Finding Formulas for Functions


Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.

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To express the side length of a square as a function of the diagonal length, start by recalling the relationship between the side length (s) and the diagonal (d) of a square. The diagonal divides the square into two right-angled triangles, where the diagonal is the hypotenuse. Using the Pythagorean theorem, we have: \( d = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2} \).
Solve for the side length (s) in terms of the diagonal (d) by rearranging the equation: \( s = \frac{d}{\sqrt{2}} \).
Now, express the area of the square as a function of the diagonal length. The area (A) of a square is given by \( A = s^2 \).
Substitute the expression for s from step 2 into the area formula: \( A = \left(\frac{d}{\sqrt{2}}\right)^2 \).
Simplify the expression for the area: \( A = \frac{d^2}{2} \). This gives the area of the square as a function of the diagonal length.

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Relationship Between Side Length and Diagonal of a Square

In a square, the relationship between the side length (s) and the diagonal (d) is defined by the Pythagorean theorem. Specifically, the diagonal can be expressed as d = s√2. This means that to find the side length as a function of the diagonal, we can rearrange this formula to s = d/√2.
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Finding Area When Bounds Are Not Given

Area of a Square

The area (A) of a square is calculated using the formula A = s², where s is the length of a side. Once we express the side length as a function of the diagonal, we can substitute this expression into the area formula to find the area in terms of the diagonal length.
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Finding Area When Bounds Are Not Given

Function Notation

Function notation is a way to represent a relationship between variables, typically written as f(x). In this context, we will express the side length and area as functions of the diagonal length, denoting them as s(d) and A(d), respectively. This notation helps clarify how one quantity depends on another.
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Solving Trigonometric Equations


For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.


sin² θ = cos² θ

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In Exercises 39–42, express the given quantity in terms of sin x and cos x.


cos (3π/2 + x)

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Finding Formulas for Functions


Consider the point (x,y) lying on the graph of y = √(x − 3). Let L be the distance between the points (x,y) and (4,0). Write L as a function of y.

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In Exercises 9–16, determine whether the function is even, odd, or neither.


𝔂 = x cos x

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Shifting Graphs


Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.


y = x³ Left 1, down 1

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General Sine Curves


For


f(x) = A sin ((2π/B)(x – C) +D


identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.


y = ½ sin (πx – x) + ½

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