Question: The displacement (in meters) of a particle
The displacement (in meters) of a particle moving in a straight line is given by s = t2 – 8t + 18, where is measured in seconds.
(a). Find the average velocity over each time interval:
(b). Find the instantaneous velocity when t = 4.
(c). Draw the graph of as a function of and draw the secant lines whose slopes are the average velocities in part (a) and the tangent line whose slope is the instantaneous velocity in part (b).
> (a). By graphing the function f (x) = (cos2x – cos x)/x2 and zooming in toward the point where the graph crosses the y-axis, estimate the value of limx→0 f (x). (b). Check your answer in part (a) by evaluating f (x) for values of that approach 0.
> Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. 9* - 5 lim -
> Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. x6 - lim 10 1-1 x" - 1 1
> Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. tan 3x lim 1-0 tan 5x
> (a). If F (x) = 5x / (1 + x2), find F' (2) and use it to find an equation of the tangent line to the curve y = 5x/ (1 + x2) at the point (2, 2). (b). Illustrate part (a) by graphing the curve and the tangent line on the same screen.
> Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. Vx + 4 – 2 lim
> (a). Show that limx→∞ 4x2 – 5x/2x2 + 1 = 2. (b). By graphing the function in part (a) and the line y = 1.9 on a common screen, find a number N such that 4x2 – 5x/2x2 + 1 > 1.9 when x > N What if 1.9 is replaced by 1.99?
> (a). Show that lim x→∞ e-x/10 = 0. (b). By graphing y = e-x/10 and y = 0.1 on a common screen, discover how large you need to make x so that e-x/10 < 0.1. (c). Can you solve part (b) without using a graphing device?
> In Chapter 7 we will be able to show, under certain assumptions, that the velocity v (t) of a falling raindrop at time t is where t is the acceleration due to gravity and is the terminal velocity of the raindrop. (a). Find limt→â
> Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). x? — 2х х 3 2.5, 2.1, 2.05, 2.01, 2.005, 2.001, lim x? — х — 2° 1.9, 1.95, 1.99, 1.995, 1.999
> Determine whether f'(0) exists.
> Determine whether f'(0) exists.
> Make a rough sketch of the curve y = xn (n an integer) for the following five cases: Then use these sketches to find the following limits.
> The quantity of oxygen that can dissolve in water depends on the temperature of the water. (So thermal pollution influences the oxygen content of water.) The graph shows how oxygen solubility varies as a function of the water temperature T. (a). What is
> By the end behavior of a function we mean the behavior of its values as x →∞ and as x →-∞. (a). Describe and compare the end behavior of the functions P (x) = 3x5 – 5x3 + 2x Q (x) 3x5 by graphing both functions in the viewing rectangles [-2, 2] by [-2, 2
> If and are continuous functions with f (3) = 5 and limx→3 [2 f (x) – g (x)] = 4, find g (3).
> Let T (t) be the temperature (in 0F) in Baltimore t hours after midnight on Sept. 26, 2007. The table shows values of this function recorded every two hours. What is the meaning of T' (10)? Estimate its value.
> Show by means of an example that limx→a [f (x) + g (x)] may exist even though neither limx→a f (x) nor limx→a g (x) exists.
> Show by means of an example that limx→a [f (x) + g (x)] may exist even though neither limx→a f (x) nor limx→a g (x) exists.
> If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli’s Law gives the volume V of water remaining in the tank after minutes as Find the rate at which the wat
> The cost (in dollars) of producing units of a certain commodity is C (x) = 5000 + 100x + 0.05x. (a) Find the average rate of change of C with respect to when the production level is changed (b). Find the instantaneous rate of change of C with respect t
> The number N of locations of a popular coffeehouse chain is given in the table. (The numbers of locations as of June 30 are given.) (a). Find the average rate of growth (i). from 2005 to 2007 (ii). from 2005 to 2006 (iii). from 2004 to 2005 In each ca
> (a). Estimate the value of lim x→∞ (√x2 + x + 1) by graphing the function f (x) = √x2 + x + 1 + x. (b). Use a table of values of f (x) to guess the value of the limit. (c). Prove that your guess is correct.
> A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F. The graph shows how the temperature of the turkey decreases and eventually approaches room
> A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a function of time. Is the initial rate of change of temperature greater or less than the rate of change after an hour?
> A particle moves along a straight line with equation of motion s = f (t), where is measured in meters and in seconds. Find the velocity and the speed when t = 5. f (t) = t-1 - t
> Explain in your own words what is meant by the equation limx→2 = f (x) = 5 Is it possible for this statement to be true and yet f (2) = 3? Explain.
> A particle moves along a straight line with equation of motion s = f (t), where is measured in meters and in seconds. Find the velocity and the speed when t = 5. f (t) = 100 + 50t - 4.9t2
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case.
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case.
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case.
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case.
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case.
> Find f' (a). f (x) = 4/ √1 - x
> Find f' (a). f (x) = √1 - 2x
> Find f' (a). f (x) = x-2
> Find f' (a). f (t) = 2t3 + t
> Sketch the graph of a function for which g (0) = g (2) = g (4) = 0, g' (1) = g' (3) = 0 g' (0) = g' (4) = 1
> Find f'(a). f (x) = 3x2 - 4x + 1
> Find the limit. limx→∞ x + 2/ √9x2 + 1)
> If g (x) = x4 - 2, find g' (1) and use it to find an equation of the tangent line to the curve y = x4 – 2 at the point (1, -1).
> If f (x) = 3x2 – x3, find f' (1) and use it to find an equation of the tangent line to the curve y = 3x2 – x3 at the point (1, 2).
> Find the limit. limx→∞ 3x + 5/ x – 4
> Find the limit. limx→2π- x csc x
> If the tangent line to y = f (x) at (4, 3) passes through the point (0, 2), find f (4) and f' (4).
> If an equation of the tangent line to the curve y = f (x) at the point where a = 2 is y = 4x - 5, find f (2) and f'(2).
> Find an equation of the tangent line to the graph of y = g (x) at x = 5 if g (5) = -3 and g'(5) = 4.
> For the function t whose graph is given, arrange the following numbers in increasing order and explain your reasoning:
> Sketch the graph of a function f for which f (0) = 0, f' (0) = 3, f'(1) = 0, and f' (2) = -1.
> The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s = 1/t2, where t is measured in seconds. Find the velocity of the particle at times t = a, t = 1, t = 2, and t = 3.
> (a). Use a graph of f (x) = (1-2/x)x to estimate the value of limx→∞ f (x) correct to two decimal places. (b). Use a table of values of f (x) to estimate the limit to four decimal places.
> If a ball is thrown into the air with a velocity of 40 ft/s, its height (in feet) after t seconds is given by y = 40t – 16t2. Find the velocity when t = 2.
> Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.
> Evaluate the limit, if it exists. lim x → 5 x2 - 5x + 6/ x – 5
> (a). Find the slope of the tangent to the curve y = 1/√x at the point where x = a. (b). Find equations of the tangent lines at the points (1, 1) and (4, ½). (c). Graph the curve and both tangents on a common screen.
> (a). Find the slope of the tangent to the curve y = 3 + 4x2 - 2x3 at the point where x = a. (b) Find equations of the tangent lines at the points (1, 5) and (2, 3). (c). Graph the curve and both tangents on a common screen.
> Find an equation of the tangent line to the curve at the given point. y = 2x + 1/x + 2, (1, 1)
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Determine lim x→1- 1/x3 – 1 and limit x→1+ 1/x3 – 1 (a). by evaluating f (x) = 1/ (x3 – 1) for values of that approach 1 from the left and from the right, (b). by reasoning as in Example 1, and (c). from a graph of f.
> (a). Can the graph of y = f (x) intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs. (b). How many horizontal asymptotes can the graph of y = f (x) have? Sketch graphs to illustrate the possibilities.
> Find an equation of the tangent line to the curve at the given point. y = x3 - 3x + 1, (2, 3)
> Find an equation of the tangent line to the curve at the given point. y = 4x - 3x2 (2, -4)
> For the function whose graph is given, state the following. (a). lim x→∞ g (x) (b). lim x→-∞ g (x) (c). lim x→3 g (x) (d). lim x→0 g (x) (e). l
> For the function f whose graph is given, state the following. (a). limx→2 f (x) (b). limx→-1- f (x) (c). limx→-1+ f (x) (d). limx→∞ f (x) (e). lim xâ†
> Graph the curve y = ex in the viewing rectangles [-1, 1] by [0, 2], [-0.5, 0.5] by [0.5, 0.5], and [-0.1, 0.1] by [0.9, 1.1. What do you notice about the curve as you zoom in toward the point (0, 1)?
> A curve has equation y = f (x). (a). Write an expression for the slope of the secant line through the points P (3, f (3)) and Q (x, f (x). (b). Write an expression for the slope of the tangent line at P.
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> In the theory of relativity, the mass of a particle with velocity is where m0 is the mass of the particle at rest and is the speed of light. What happens as v→c-?
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.
> Find limx→∞ f (x) if, for all x > 1,
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> To prove that sine is continuous we need to show that limx→a sin x = sin a for every real number a. If we let h = x - a, then x = a + h and x→a ⇔ h → 0. So, an equivalent statement is that limh→a sin (a + h) = sin a Use (6) to show that this is true.
> If limx→1 f (x)/x2 = 5 find the following limits. (a). limx→0 f (x) (b). limx→0 f (x)/x
> Estimate the horizontal asymptote of the function f (x) = 3x2 + 500 x2/ x2 + 500 x2 + 100x + 2000 by graphing f for -10 < x < 10. Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?
> (a). Use a graph of f (x) = √3x2 + 8x + 6 – 3x2 + 3x + 1 to estimate the value of limx→∞ f (x) to one decimal place. (b). Use a table of values of f (x) to estimate the limit to four decimal places. (c). Find the exact value of the limit.
> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.
> If f (x) = [[x]] + [[-x]], show that limx→2 f (x) exists but is not equal to f (2).
> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.
> Guess the value of the limit x→∞ x2/2x by evaluating the function f (x) = x2/2x for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of f to support your guess.
> Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.
> (a). Graph the function f (x) = √2x2 + 1/3x - 5 How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits lim (b). By calculating values of f, give numerical estimates of the limits
> Find the limit. limx→∞ x + x3 + x5/1 – x2 + x4
> Find the limit. limx→(π/2)+ etanx
> Find the limit. limx→-∞ (x4 + x5)
> Find the limit. limx→∞ e3x – e-3x/ e3x + e-3x
> Find the limit. limx→∞ (e-2x cos x)
> Find the limit. limx→∞ sin2x/x2
> Find the limit. limx→∞ cos x
> Find the limit. limx→∞ e√x2 + 1
> A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. (a). If P is the point (15, 250) on the graph of V, fin
> Find the limit. limx→∞ e-x2
> Find the limit. limx→∞ (√x2 + ax – √x2 + bx
> Find the limit. limx→∞ (√9x2 + x – 3x)
> Find the limit. Limu→∞ 4u4 + 5/ (u2 – 2) (2u2 – 1)
> Find the limit. limx→-∞ t2 + 2/ t3 + t2 – 1
> Find the limit. limx→∞ x3 + 5x/ 2x3 – x2 + 4
> Find the limit. limx→2- x2 – 2x/ (x2 – 4x + 4
