Questions and Answers of Calculus Graphical, Numerical, Algebraic

Find the instantaneous rate of change of the surface area S = 6x2 of a cube with respect to the edge length x at x = a.
In Exercises the curve y = ∫(x) has a vertical tangent at x = a ifIn each case, the right- and left-hand limits are required to be the same: both + ∞ or both - ∞. Use graphs to investigate
Is any real number exactly 1 less than its fourth power? Give any such values accurate to 3 decimal places.
In Exercises find (a) a simple basic function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function.y = x2 + e-x
Which of the following points of discontinuity ofis not removable?(A) x = - 1(B) x = 0(C) x = 1(D) x = 2(E) x = 3 f(x) = x(x - 1)(x - 2)(x + 1)(x - 3) x(x - 1)(x-2)(x + 1)(x-3)
True or False It is possible to extend the definition of a function ∫ at a jump discontinuity x = a so that ∫ is continuous at x = a. Justify your answer.
In Exercises find the limit graphically. Use the Sandwich Theorem to confirm your answer. lim x² sin x X-0
(a) Find the domain of ∫. (b) Draw the graph of ∫.(c) Explain why x = - 1 and x = 0 are points of discontinuity of ∫.(d) Are either of the discontinuities in part (c) removable?
True or False It is possible for a function to have more than one horizontal asymptote. Justify your answer.
In Exercises find the limit graphically. Use the Sandwich Theorem to confirm your answer. lim x² sin
Show that ∫(x) is continuous at x = a if and only if lim f(a+h) = f(a). h→0
In Exercises find the limit graphically. Use the Sandwich Theorem to confirm your answer. lim x² cos X-0 -12
Give a convincing argument that the following function is not continuous at any real number. f(x) = [1, 10, if x is rational if x is irrational
What is the value of (A) 5/2 (B) 3/2 (C) 1 (D) 0 (E) Does not exist lim f(x)?
A rock released from rest to fall on a small airless planet falls y = gt2 m in t sec, g a constant. Suppose that the rock falls to the bottom of a crevasse 20 m below and reaches the bottom in 4
In Exercises find the limit. Give a convincing argument that the value is correct. In x² lim x In x
What is the value of ∫(1)?(A) 5/2 (B) 3/2 (C) 1 (D) 0 (E) Does not exist
In Exercises complete the following tables and state what you believe limx→0 ∫(x) to be.a.b. X f(x) -0.1 -0.01 ? ? -0.001 ? -0.0001
In Exercises (a) find each point of discontinuity, (b) Which of the discontinuities are removable? not removable? Give reasons for your answers. -1 2 0 1 y = f(x) 2 3 X
In Exercises determine the limit graphically. Confirm algebraically. lim X-0 sin 2x X
In Exercises find limx→∞ y and limx→ -∞ y. y = cos (1/x) 1 + (1/x)
In Exercises determine the limit graphically. Confirm algebraically. lim x→0 (2 + x)³-8 X
In Exercises (a) find each point of discontinuity, (b) Which of the discontinuities are removable? not removable? Give reasons for your answers. -1 0 y = f(x) 2 X
An object is dropped from the top of a 100-m tower. Its height above ground after t sec is 100 - 4.9t2 m. How fast is it falling 2 sec after it is dropped?
In Exercises find limx→∞ y and limx→ -∞ y. y X + 5x²1 x²
In Exercises (a) find each point of discontinuity, (b) Which of the discontinuities are removable? not removable? Give reasons for your answers. - f(x) = (1-x², x-1 12. x = -1
In Exercises (a) find the slope of the curve at x = a. (b) Describe what happens to the tangent at x = a as a changes. y= 1 x-1
In Exercises (a) find the slope of the curve at x = a. (b) Describe what happens to the tangent at x = a as a changes.y = 9 - x2
In Exercises determine the limit graphically. Confirm algebraically. lim X-0 1 2+x X 1 2
In Exercises find limx→∞ y and limx→ -∞ y. y 5-12- x2 X x+15+x²
In Exercises determine the limit graphically. Confirm algebraically. 5x³ + 8x² lim 1-0 3x4-16x²
In Exercises (a) find each point of discontinuity, (b) Which of the discontinuities are removable? not removable? Give reasons for your answers. 1 f(x)=x-1' x³ - 2x + 5, x < 1 x≥1
In Exercises use graphs and tables to find the limits. lim sec x (m/2)¹
In Exercises (a) find the slope of the curve at x = a. (b) Describe what happens to the tangent at x = a as a changes.y = 2/x
In Exercises (a) find each point of discontinuity, (b) Which of the discontinuities are removable? not removable? Give reasons for your answers. 3-x. x/2, f(x) = {2, x < 2 x = 2 x>2
In Exercises determine whether the limit exists on the basis of the graph of y = ∫(x). The domain of ∫ is the set of real numbers. lim f(x)
In Exercises determine the limit graphically. Confirm algebraically. lim 12-31 +2 12-4
In Exercises use graphs and tables to find the limits. lim csc x 3-0+
In Exercises (a) find the slope of the curve at x = a. (b) Describe what happens to the tangent at x = a as a changes.y = x2 + 2
In Exercises (a) find each point of discontinuity, (b) Which of the discontinuities are removable? not removable? Give reasons for your answers. (3-x, f(x) = x 2 + 1, x2
In Exercises determine the limit graphically. Confirm algebraically. x-1 lim X1 X*
In Exercises determine whether the limit exists on the basis of the graph of y = ∫(x). The domain of ∫ is the set of real numbers. lim f(x) ab
In Exercises use graphs and tables to find the limits. lim int x X
In Exercises determine whether the curve has a tangent at the indicated point. If it does, give its slope. If not, explain why not. f(x) sinx, cos x, 0≤x≤ 3π/4 37/4 ≤x≤ 2π at x = 37/4
In Exercises use the function ∫ defined and graphed below to answer the questions.Is it possible to extend ∫ to be continuous at x = 3? If so, what value should the extended function have there?
In Exercises determine whether the limit exists on the basis of the graph of y = ∫(x). The domain of ∫ is the set of real numbers. lim f(x) X-C
In Exercises explain why you cannot use substitution to determine the limit. Find the limit if it exists. lim 1-0 (4 + x)²-16 X
In Exercises use graphs and tables to find the limits. lim x-01 int x X
In Exercises a distance-time graph is shown.(a) Estimate the slopes of the secants PQ1, PQ2, PQ3, and PQ4, arranging them in order in a table. What is the appropriate unit for these slopes?(b)
In Exercises find the remaining functions in the list of functions: ∫, g, ∫ o g , g o ∫ f(x): 2x - 1 x+5 g(x) = 1 X +1
In Exercises find the limits. lim √9-x² X-5
Find each limit. Let f(x) = [x² - 4x + 5, (4-x, x
Which of the following lines is a horizontal asymptote for f(x): = 3xx+x-7, 2x + 4x-5 -?
In Exercises use the vertical line test (see Exercise 35) to determine whether the curve is the graph of a function.
In Exercises determine the limit. Assume that lim f(x) = 7 and lim g(x) = -3. x-b (a) lim (f(x) + g(x)) (b) lim (f(x) g(x)) 1-b (c) lim 4 g(x) f(x) (d) lim x→h x-b g(x)
The number of bears in a federal wildlife reserve is given by the population equationwhere t is in years.(a) Find p (0). Give a possible interpretation of this number.(b) Find(c) Give a
Which of the following is an end behavior for(A) x3 (B) 2x3 (C) 1/x3 (D) 2 (E) 1/2 f(x) = 2₁³-x²+x+1₂ ³-1
True or False If ∫(x) has a vertical asymptote at x = c, then either limx→c- ∫(x) = limx→c+ ∫(x) = ∞ or limx→c- ∫(x) =limx→c+ ∫(x) = - ∞. Justify your answer.
Let ∫ be continuous and never zero on [a, b]. Show that either ∫(x) > 0 for all x in [a, b] or ∫(x) < 0 for all x in [a, b],
A water balloon dropped from a window high above the ground falls y = 4.9t2 m in t sec. Find the balloon’s(a) Average speed during the first 3 sec of fall.(b) Speed at the instant t = 3.
Prove that if ∫ is continuous on an interval, then so is |∫|.
In Exercises find the limit. Give a convincing argument that the value is correct. In x lim x→log.x
What is the value of limx→1+ ∫(x)?(A) 5/2 (B) 3/2 (C) 1 (D) 0 (E) Does not exist
Explain why there is no value L for which limx→∞ sin x = L.
In Exercises find the limit. Give a convincing argument that the value is correct. lim X- In (x + 1) In x
What is the value of limx→1 ∫(x)?(A) 5/2 (B) 3/2 (C) 1 (D) 0 (E) Does not exist
In Exercises complete the following tables and state what you believe limx→0 ∫(x) to be. (a) X f(x) (b) f(x) -0.1 -0.01 -0.001 -0.0001 ? ? 0.1 0.01 0.001 0.0001 ? ? ? ?
In Exercises complete the following tables and state what you believe limx→0 ∫(x) to be. (a) X f(x) (b) f(x) -0.1 -0.01 -0.001 -0.0001 ? ? 0.1 0.01 0.001 0.0001 ? ? ? ?
In Exercises complete the following tables and state what you believe limx→0 ∫(x) to be. (a) X f(x) (b) f(x) -0.1 -0.01 -0.001 -0.0001 ? ? 0.1 0.01 0.001 0.0001 ? ? ? ?
To prove that limθ→0 (sin θ)/θ = 1 when θ is measured in radians, the plan is to show that the right- and left-hand limits are both 1.(a) To show that the right-hand limit is 1, explain why we
In Exercises match the function with the table. Y₁ x²-x-2 x-1
Find the slope of the curve y = x2 - x - 2 at x = a.
In Exercises the curve y = ∫(x) has a vertical tangent at x = a ifIn each case, the right- and left-hand limits are required to be the same: both + ∞ or both - ∞. Use graphs to investigate
Is any real number exactly 2 more than its cube? Give any such values accurate to 3 decimal places.
In Exercises find (a) a simple basic function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function.y = x + In |x|
Let ∫(x) = x2 - 3x and P = (1, ∫(1)). Find (a) The slope of the curve y = ∫(x) at P, (b) An equation of the tangent at P, (c) An equation of the normal at P.
In Exercises match the function with the table. x2 - 2x + 1 x-1
Find a value for a so that the functionIs continuous. f(x)= x² -1, 2ax, x
In Exercises determine whether the graph of the function has a tangent at the origin. Explain your answer. f(x) = x² sin 0, x #0 x=0
In Exercises match the function with the table. Y₁ = x²+x-2 x + 1
In Exercises find (a) a simple basic function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function.y = x2 + sin x
At what points, if any, are the tangents to the graph of ∫(x) = x2 — 3x horizontal?
Find a value for a so that the functionIs continuous. f(x) = 2x + 3, lar + 1, ax x≤2 x>2
In Exercises determine whether the graph of the function has a tangent at the origin. Explain your answer. f(x) x sin 0, 1 X x #0 x=0
In Exercises use the graph of y = ∫(1/x) to find limx→∞ ∫(x) and limx→-∞ ∫(x).∫(x) = xex
Find a value for a so that the functionIS continuous. f(x)= 14-x², x < -1 lax2-1,-1
Estimate the slope of the curve y = sin x at x = 1.
In Exercises use the graph of y = ∫(1/x) to find limx→∞ ∫(x) and limx→-∞ ∫(x).∫(x) = x2e-x
Bluetop Cab charges $3.20 for the first mile and $1.35 for each additional mile or part of a mile.(a) Write a formula that gives the charge for x miles with 0 ≤ x ≤ 20.(b) Graph the function in
Find a value for a so that the functionIS continuous. x² + x + a₂ x < 1 2 x 21 f(x) = {x² +
In Exercises use the graph of y = ∫(1/x) to find limx→∞ ∫(x) and limx→-∞ ∫(x). f(x) = In x X
Table 2.4 gives the population of Florida for several years.(a) Let x = 0 represent 1990, x = I represent 1991, and so forth. Make a scatter plot for the data.(b) Let P represent the point
In Exercises complete parts (a), (b), and (c) for the piecewise defined function.(a) Draw the graph of ∫.(b) Determine limx→c+ ∫(x) and limx→c- ∫(x).(c) Does limx→c ∫(x ) exist? If
Explain why the equation e-x = x has at least one solution.
In Exercises complete parts (a), (b), and (c) for the piecewise defined function.(a) Draw the graph of ∫.(b) Determine limx→c+ ∫(x) and limx→c- ∫(x).(c) Does limx→c ∫(x ) exist? If
In Exercises use the graph of y = ∫(1/x) to find limx→∞ ∫(x) and limx→-∞ ∫(x). f(x) = x sin
A welder’s contract promises a 3.5% salary increase each year for 4 years and Luisa has an initial salary of $36,500.(a) Show that Luisa’s salary is given bywhere t is the time, measured in
(a) Find the domain of ∫.(b) Write an equation for each vertical asymptote of the graph of ∫.(c) Write an equation for each horizontal asymptote of the graph of ∫.(d) Is ∫ odd, even, or
In Exercises complete parts (a), (b), and (c) for the piecewise defined function.(a) Draw the graph of ∫.(b) Determine limx→c+ ∫(x) and limx→c- ∫(x).(c) Does limx→c ∫(x ) exist? If
In Exercises find the limit of ∫(x) as (a) x→ - ∞, (b) x→∞, (c) x→0-, and (d) x→0+. f(x) = (1/x, x

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