Questions and Answers of Calculus Graphical, Numerical, Algebraic

In Exercises explain why you cannot use substitution to determine the limit. Find the limit if it exists. lim x-0 X
In Exercises use limx→c k = k, limx→c x = c, and the properties of limits to find the limit. lim (2x³ 3x²+x-1)
In Exercises use graphs and tables to find (a) limx→∞ ∫(x) and (b) limx→ -∞ ∫(x) (c) Identify all horizontal asymptotes. f(x)= 3x³x+1 x + 3
In Exercises find the slope of the line determined by the points.(- 3 , -1), (3, 3)
In Exercises find the average rate of change of the function over each interval.∫(x) = In x(a) [1, 4] (b) [100, 103]
In Exercises find ∫(2). f(x) = 3x - 1. I ²-1' x
In Exercises an object dropped from rest from the top of a tall building falls y = 16t2 feet in the first t seconds.Find the speed of the object at t = 4 seconds and confirm your answer algebraically.
In Exercises find the limits. lim -0 1 2+x X 2
In Exercises find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. y = √2x + 3
In Exercises find the quotient q(x) and remainder r(x) when ∫(x) is divided by g(x).∫(x) = 2x3 - 3x2 + x - 1, g(x) = 3x3 + 4x - 5
In Exercises find the average rate of change of the function over each interval.∫(x) = cot t(a) [π/4, 3π/4] (b) [π/ 6, π/2]
In Exercises use graphs and tables to find (a) limx→∞ ∫(x) and (b) limx→ -∞ ∫(x) (c) Identify all horizontal asymptotes. . f(x) = 3x + 1 [x] + 2
In Exercises write the inequality in the form a < x < b|x| < 4
In Exercises find the limits. 2x² + 3 lim -±= 5x² + 7
Let y1 = In (x/a), y2 = In x, y3 = y2 - yt, and y4 = ey3.(a) Graph y1 and y2 for a = 2,3, 4, and 5. How are the graphs of y1 and y2 related?(b) Graph y3 for a = 2,3, 4, and 5. Describe the
True or False The function ∫(x) = x-3 is an odd function. Justify your answer.
Let y = a sin x + b cos x.Use the symbolic manipulator of a computer algebra system (CAS) to help you with the following:(a) Express y as a sinusoid for the following pairs of values(b) Conjecture
Solve the equation sin x = - 0.2 in the following intervals.(a) 0 ≤ x < 2π(b) - ∞ < x < ∞
Which of the following gives the domain of(A) x ≠ ±3(B) (-3, 3)(C) [-3, 3](D) (-∞, -3) ∪ (3, ∞)(E) (3, ∞) f(x) = X V9 - x²
Solve for x: e-0.2x = 4
If ∫ is a one-to-one function, prove that g(x) = -∫ (x) is also one-to-one.
In Exercises show that the function is periodic and find its period.y = sin3 x
The graph of / is shown. Draw the graph of each function.(a) y = ∫(- x)(b) y = - ∫(x)(c) y = - 2∫(x+ 1) +1(d) y = 3∫(x - 2) - 2 -1
Which of the following gives the range of(A) (- ∞, 1) ∪ (1, ∞) (B) x ≠ l(C) All real numbers(D) (-∞, 0) ∪ (0, ∞) (E) x ≠ 0 f(x) = 1 + x-1
If ∫ is a one-to-one function and ∫(x) is never zero, prove that g(x) = 1/ ∫(x) is also one-to-one.
A portion of the graph of a function defined on [- 3 , 3] is shown. Complete the graph assuming that the function is(a) Even.(b) Odd. 2 1 -1 (1.-1) (3, 2) 2 3 X
In Exercises show that the function is periodic and find its period.y = |tan x|
If ∫(x) = 2x - 1 and g(x) = x + 3, which of the following gives (∫ o g)(2)?(A) 2 (B) 6 (C) 7 (D) 9 (E) 10
Suppose that a ≠ 0, b ≠ 1, and b > 0. Determine the domain and range of the function.(a) y = a(bc-x) + d (b) y = a logb (x - c) + d
In Exercises graph one period of the function.∫(x) = sin (60x)
Smith Hauling purchased an 18-wheel truck for $100,000. The truck depreciates at the constant rate of $10,000 per year for 10 years.(a) Write an expression that gives the value y after x years.(b)
The length L of a rectangle is twice as long as its width W. Which of the following gives the area A of the rectangle as a function of its width?(A) A(W) = 3W(B) A(W) = 1/2 W2(C) A(W) = 2 W2(D) A(W)
(a) Give a convincing argument that ∫ is one-to-one.(b) Find a formula for the inverse of ∫.(c) Find the horizontal and vertical asymptotes of ∫.(d) Find the horizontal and vertical asymptotes
If Joenita invests $1500 in a retirement account that earns 8% compounded annually, how long will it take this single payment to grow to $5000?
In Exercises (a) graph ∫ o g and g o ∫ and make a conjecture about the domain and range of each function, (b) Then confirm your conjectures by finding formulas for∫ o g and g o ∫.∫(x) = x -
A drug is administered intravenously for pain. The functiongives the number of units of the drug in the body after t hours.(a) What was the initial number of units of the drug administered?(b) How
In Exercises graph one period of the function.∫(x) = cos (60πx)
In Exercises (a) graph ∫ o g and g o ∫ and make a conjecture about the domain and range of each function, (b) Then confirm your conjectures by finding formulas for∫ o g and g o ∫.∫(x) = 1 -
The number of guppies in Susan’s aquarium doubles every day. There are four guppies initially.(a) Write the number of guppies as a function of time t.(b) How many guppies were present after 4 days?
Table 1.23 shows the number of doctoral degrees earned by Hispanic students for several years. Let x = 0 represent 1980, x = 1 represent 1981, and so forth.(a) Find a linear regression equation for
In Exercises (a) graph ∫ o g and g o ∫ and make a conjecture about the domain and range of each function, (b) Then confirm your conjectures by finding formulas for∫ o g and g o ∫.∫(x) = x2
In Exercises (a) graph ∫ o g and g o ∫ and make a conjecture about the domain and range of each function, (b) Then confirm your conjectures by finding formulas for∫ o g and g o ∫.
Table 1.24 shows the population of New York State for several years. Let x = 0 represent 1980, x = 1 represent 1981, and so forth.(a) Find the exponential regression equation for the data and
In Exercises a portion of the graph of a function defined on [-2, 2] is shown. Complete each graph assuming that the graph is (a) even, (b) odd. 1.5 0 y=f(x) 1 2 X
Consider the point P(- 2, 1) and the line L: x + y = 2.(a) Find the slope of L.(b) Write an equation for the line through P and parallel to L.(c) Write an equation for the line through P and
In Exercises a portion of the graph of a function defined on [-2, 2] is shown. Complete each graph assuming that the graph is (a) even, (b) odd. 0 2 X
Let ∫(x) = 1 - ln (x - 2).(a) What is the domain of ∫? (b) What is the range of ∫?(c) What are the x-intercepts of the graph of ∫?(d) Find ∫-1 (e) Confirm your answer
In Exercises a portion of the graph of a function defined on [-2, 2] is shown. Complete each graph assuming that the graph is (a) even, (b) odd. -2 1.3 0 X
Which of the following values is the average rate of ∫(x) = √x + 1 over the interval (0, 3)?(A) - 3 (B) - 1 (C) -1/3 (D) 1/3 (E) 3
(a) Must the product of two even functions always be even? Give reasons for your answer.(b) Can anything be said about the product of two odd functions? Give reasons for your answer.
Enter y1= √x, y2 = √1 - x and y3 = y1 + y2 on your grapher.(a) Graph y3 in [-3, 3] by [-1, 3].(b) Compare the domain of the graph of y3 with the domains of the graphs of y1 and y2.(c)
Table 1.22 gives the values of the function ∫(x) = a sin
In Exercises let x = 0 represent 1990, x = 1 represent 1991, and so forth.(a) Find a natural logarithm regression equation for the data in Table 1.17 and superimpose its graph on a scatter plot of
In Exercises 49, assume that the graph of the exponential function f(x) = k- ax passes through the two points. Find the values of a and k.(1, 1.5), ( - 1 , 6)
In Exercises give a parametrization for the curve.The line segment with endpoints (- 2, 5) and (4, 3)
In Exercises (a) draw the graph of the function. Then find its (b) domain and (c) range.∫(x) = 2|x + 4| - 3
Which of the following is an equation of the vertical line through (-2, 4)?(A) y = 4 (B) x = 2(C) y = - 4(D) x = 0 (E) x = - 2
True or False The period of y = sin (x/2) is π. Justify your answer.
Which of the following is an equation of the line through (- 2, - 1) parallel to the line y = - 3x + 1?(A) y = -3x + 5(B) y = -3x - 7 (C)(D) y = -3x + 1(E) y = -3x - 4 y= 3 3
In Exercises give a parametrization for the curve.The line through ( - 3, - 2) and (4, - 1)
Let y = ∫(x) = mx + b, m ≠ 0.(a) Give a convincing argument that ∫ is a one-to-one function.(b) Find a formula for the inverse of ∫. How are the slopes of ∫ and ∫-1 related?(c) If the
True or False The function displayed in the graph below is one-to-one. Justify your answer. x
In Exercises find(a) ∫(g(x)) (b) g(∫(x)) (c) ∫(g(0))(d) g(∫(0)) (e) g(g(-2)) (f) ∫(∫(x))f(x) = x + 5, g(x) = x2 - 3
The median price of existing single-family homes has increased consistently during the past few years. However, the data in Table 1.4 show that there have been differences in various parts of the
Which of the following is the x-intercept of the line y = 2x -5?(A) x = - 5 (B) x = 5 (C) x = 0(D) x = 5/2 (E) x = - 5/2
True or False The amplitude of y = 1/2 cos x is 1. Justify your answer.
In Exercises give a parametrization for the curve.The ray with initial point (2, 5) that passes through (- 1, 0)
Copy and complete the following table. (a) (b) (c) (d) g(x) ? ? 1/x Vx f(x) Vx-5 1+1/x ? ? (fog)(x) Vx²-5 X [x], x ≥ 0
Table 1.5 shows the gross revenue for the Broadway season in millions of dollars for several years.(a) Find the quadratic regression for the data in Table 1.5. Let x = 0 represent 1990, x = 1
In Exercises find(a) ∫(g(x)) (b) g(∫(x)) (c) ∫(g(0))(d) g(∫(0)) (e) g(g(-2)) (f) ∫(∫(x))f(x) = x + 1, g(x) = X - 1
In Exercises ∫(x) = 2 cos(4x + π ) - 1.Which of the following is the domain of ∫?(A) [-π, π] (B) [-3, 1] (C) [-1, 4](D) (-∞, ∞) (E) X ≠ 0
In Exercises give a parametrization for the curve.y = x (x - 4), x ≤ 2
True or False f (∫ º g)(x) = x, then g is the inverse function off. Justify your answer.
In Exercises ∫(x) = 2 cos(4x + π ) - 1.Which of the following is the range of ∫?(A) (-3 ,1)(B) [-3, 1](C) (-1, 4)(D) [-1, 4](E) (-∞, ∞)
In Exercises do the following.(a) Find ∫-1 and show that (∫ o ∫-1) (x) = (∫-1 o ∫)(x) = x.(b) Graph ∫ and ∫-1 in the same viewing window.∫(x) = 2 - 3x
In Exercises use the function ∫(x) = 3 - In (x + 2).Which of the following is the domain of ∫?(A) x ≠ - 2 (B) (- ∞, ∞) (C) (- 2 , ∞)(D) [-1.9, ∞) (E) (0, ∞)
Begin with a circular piece of paper with a 4-in. radius as shown in (a). Cut out a sector with an arc length of x. Join the two edges of the remaining portion to form a cone with radius r and height
In Exercises ∫(x) = 2 cos(4x + π ) - 1.Which of the following is the period of ∫?(A) 4π (B) 3π(C) 2π (D) π (E) π/2
In Exercises do the following.(a) Find ∫-1 and show that (∫ o ∫-1) (x) = (∫-1 o ∫)(x) = x.(b) Graph ∫ and ∫-1 in the same viewing window.∫(x) = (x + 2)2, x ≥ - 2
In Exercises use the function ∫(x) = 3 - In (x + 2).Which of the following is the range of ∫?(A) (-∞, ∞)(B) (-∞, 0)(C) (-2 , ∞)(D) (0, ∞)(E) (0, 5.3)
Which of the following is the inverse of ∫(x) = 3x - 2?(A)(B) g(x) = x(C) g(x) = 3x - 2(D) (E) g(x) = 1 3x-2
Three different parallelograms have vertices at (-1, 1), (2, 0), and (2, 3). Draw the three and give the coordinates of the missing vertices.
Dayton Power and Light, Inc., has a power plant on the Miami River where the river is 800 ft wide. To lay a new cable from the plant to a location in the city 2 mi downstream on the opposite side
In Exercises find the measure of the angle in radians and degrees.sin-1 (0.6)
Which of the following is the measure of tan-1(-√3) in degrees?(A) -60° (B) -30° (C) 30° (D) 60° (E) 120°
Let ∫(x) = sin x + cos x.(a) Graph y = ∫(x). Describe the graph.(b) Use the graph to identify the amplitude, period, horizontal shift, and vertical shift.(c) Use the formulafor the sine of the
Show that if the midpoints of consecutive sides of any quadrilateral are connected, the result is a parallelogram.
In Exercises find the measure of the angle in radians and degrees.tan-1 (- 2. 3)
Consider the circle of radius 5 centered at (0, 0). Find an equation of the line tangent to the circle at the point (3, 4).
Which of the following is a solution of the equation 2 - 3-x = -1?(A) x = -2 (B) x = - 1 (C) x = 0(D) x = 1 (E) There are no solutions.
True or False The function ∫(x) = x4 + x2 + x is an even function. Justify your answer.
Find the six trigonometric values of θ = cos-1 (3/7). Give exact answers.
In Exercises find the (a) domain and (b) range, and (c) graph the function. y=-1+√2-x
In Exercises solve the equation algebraically. Support your solution graphically.2X + 2-x = 5
In Exercises find the (a) domain and (b) range, and (c) graph the function. |V=x. -4≤x≤0 Vx, 0≤x≤4
In Exercises find the (a) domain and (b) range, and (c) graph the function. (-x-2, -2≤x≤-1 -1
In Exercises write a piecewise formula for the function. 2 (2.1) 2 5
John invests $200 at 4.5% compounded annually. About how long will it take for John’s investment to double in value?(A) 6 yrs (B) 9 yrs (C) 12 yrs (D) 16 yrs (E) 20 yrs
The pressure p experienced by a diver under water is related to the diver’s depth d by an equation of the form p = kd + 1 (k a constant). When d = 0 meters, the pressure is 1 atmosphere. The
Find the (a) Amplitude(b) Period (c) Horizontal shift (d) Vertical shift of the model used in the figure below (e) Then write the equation for the modelNormal mean air

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