Questions and Answers of Calculus 4th

Find all points in the xy-plane with integer coordinates located at a distance 5 from the origin. Then find all points with integer coordinates located at a distance 5 from (2, 3).
Give an example of a function whose domain D has three elements and whose range R has two elements. Does a function exist whose domain D has two elements and whose range R has three elements?
Find the domain and range of the function.g(t) = t4
Find the domain and range of the function.ƒ(x) = x3
Find the domain and range of the function.g(t) = √2 − t
Find the domain and range of the function.h(s) = 1/s
Find the domain and range of the function.ƒ(x) = 1x2
Find the domain and range of the function.g(t) = 1/√1 − t
Determine where ƒ is increasing.ƒ(x) = |x + 1|
Determine where ƒ is increasing.ƒ(x) = x3
Determine where ƒ is increasing.ƒ(x) = x4
Determine where ƒ is increasing.ƒ(x) = 1/x4 + x2 + 1
Find the zeros of ƒ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.ƒ(x) = x2 − 4
Find the zeros of ƒ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.ƒ(x) = 2x2 − 4
Find the zeros of ƒ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.ƒ(x) = x3 − 4x
Find the zeros of ƒ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.ƒ(x) = x3
Find the zeros of ƒ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.ƒ(x) = 2 − x3
Which of the curves in Figure 27 is the graph of a function of x? b. ➤X (A) W∞ (B) 木ヒ (D) (E) →X (C) (F) X →X
Find the zeros of ƒ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.ƒ(x) = 1/(x−1)2 + 1
Of the curves in Figure 27 that are graphs of functions, which is the graph of an odd function? Of an even function? b (A) ➤X W∞ (B) 木ヒ (D) (E) →X (C) (F) →X
Determine whether the function is even, odd, or neither.(a) ƒ(x) = x5 (b) g(t) = t3 − t2 (c) F(t) = 1/t4 + t2
Determine whether the function is even, odd, or neither. (a) f(x) = 2x-x² (c) f(t) = A +1+1 - A-1+1 (b) k(w) = (1-w)³ +(1+w)³
Write ƒ(x) = 2x4 − 5x3 + 12x2 − 3x + 4 as the sum of an even and an odd function.
Assume that p is a function that is defined for all x.(a) Prove that if ƒ is defined by ƒ(x) = p(x) + p(−x) then ƒ is even.(b) Prove that if g is defined by g(x) = p(x) − p(−x) then g is odd.
Let ƒ be the function shown in Figure 28.Extend the graph of f to [−4, 4] so that it is an odd function. 3 نا 2 1 0 y 1 2 3 FIGURE 28 4 -X
Suppose that ƒ has domain [4, 8] and range [2, 6]. Find the domain and range of:(a) y = ƒ(x) + 3 (b) y = ƒ(x + 3)(c) y = ƒ(3x) (d) y = 3 ƒ(x)
Figure 29 shows the graph of ƒ(x) = |x| + 1. Match the functions (a)–(e) with their graphs (i)–(v). (a) y = f(x - 1) (d) y = f(x-1)-2 (b) y = -f(x) (e) y = f(x + 1) (c) y =-f(x) + 2
Let ƒ(x) = x2. Sketch the graph over [−2, 2] of:(a) y = ƒ(x + 1) (b) y = ƒ(x) + 1(c) y = ƒ(5x)(d) y = 5ƒ(x)
Suppose that the graph of ƒ(x) = x4 − x2 is compressed horizontally by a factor of 2 and then shifted 5 units to the right.(a) What is the equation for the new graph?(b) What is the equation if
Sketch the graph of y = ƒ(2x) and y =ƒ(1/2x), where ƒ(x) = |x| + 1 (Figure 29). V 3- 2+ -3-2-1 -1 -2. y = f(x) = lxl + 1 3 2 -1 + 1 (ii) 2 3 1 처 ㅎㅎ (i) 3. 2+ 1 (iv) + + 2 3 + + 1 2
Find the function ƒ whose graph is obtained by shifting the parabola y = x2 by 3 units to the right and 4 units down, as in Figure 30. y y = x2 ку 3 T y = f(x)
For each curve in Figure 31, state whether it is symmetric with respect to the y-axis, the origin, both, or neither. y (A) y £ (B) می شهر مرند (C) y (D) **
Define ƒ(x) to be the larger of x and 2 − x. Sketch the graph of ƒ. What are its domain and range? Express ƒ. (x) in terms of the absolute value function.
Suppose that ƒ and g are both odd. Which of the following functions are even? Which are odd? (a) y = f(x)g(x) (c) y = f(x) - g(x) (b) y (d) y = = f(x)³ f(x) g(x)
Prove the triangle inequality (|a + b| ≤ |a| + |b|) by adding the two inequalities: -lal ≤ a ≤ lal, -|b1 ≤ b ≤|bl
Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only ifObserve that r has a finite decimal expansion when 10Nr is an integer for some N ≥ 0 (and hence b divides
Let p = p1 . . . ps be an integer with digits p1, . . . , ps. Show thatUse this to find the decimal expansion of r = 2/11. Note that P 10s 1 = 0.p1... Ps
Formulate a condition for ƒ to be symmetric with respect to the point (a, 0) on the x-axis.
Let ƒ be the function shown in Figure 28.Extend the graph of ƒ to [−4, 4] so that it is an even function. 3 نا 2 1 0 y 1 2 3 FIGURE 28 4 -X
Let ƒ be the function shown in Figure 28.Sketch the graphs of y = ƒ(−x) and y = −ƒ(−x). 3 نا 2 1 0 y 1 2 3 FIGURE 28 4 -X
Let ƒ be the function shown in Figure 28.Sketch the graphs of y = ƒ(2x), y = ƒ(1/2x), and y = 2ƒ(x). 3 نا 2 1 0 y 1 2 3 FIGURE 28 4 -X
Let ƒ be the function shown in Figure 28.Sketch the graphs of y = ƒ(x + 2) and y = ƒ(x) + 2. 3 نا 2 1 0 y 1 2 3 FIGURE 28 4 -X
State whether the function is increasing, decreasing, or neither.(a) Surface area of a sphere as a function of its radius(b) Temperature at a point on the equator as a function of time(c) Price of an
Let ƒ be the function shown in Figure 28.Find the domain and range of ƒ. 3 نا 2 1 0 y 1 2 3 FIGURE 28 4 -X

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