(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?
> (a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root correct to three decimal places. 100e-x/100 = 0.01x2
> (a). Prove that the equation has at least one real root. (b). Use your calculator to find an interval of length 0.01 that contains a root. ln x = 3 – 2x
> Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. x4 + x - 3 = 0 (1, 2)
> Let f (x) = [[cos x]], -Ï€ (a). Sketch the graph of (b). Evaluate each limit, if it exists. (c). For what values of a does limx→ a f (x) exist? (i) lim f(x) (ii) lịm f(x) I-(m/2)- (iii) lim f(x) 1-(m/2)+ (iv) lim f(x) 1/
> (a). A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Show that the concentration of salt after t minutes (in grams per liter) is (b). What happens to the concentra
> (a). Show that the absolute value function F (x) = |x| is continuous everywhere. (b). Prove that if f is a continuous function on an interval, then so is |f|. (c). Is the converse of the statement in part (b) also true? In other words, if |f| is continuo
> Show that the function is continuous on (-∞, ∞). [x* sin(1/x) if x # 0 f(x) = if x= 0
> The graph shows the influence of the temperature T on the maximum sustainable swimming speed of Coho salmon. (a). What is the meaning of the derivative S'(T)? What are its units? (b). Estimate the values of S'(15) and S'(25) and interpret them. SA
> Let P and Q be polynomials. Find limx→∞ P (x)/Q (x) if the degree of P is (a) less than the degree of Q and (b) greater than the degree of Q.
> The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is Q = f (p). (a). What is the meaning of the derivative f' (8)? What are its units? (b). Is f'(8) positive or negative? Explain.
> Use the given graph of f to state the value of each quantity, if it exists. If it does not exist, explain why. 2- 4 2. 4, (a) lim f(x) (b) lim f(x) (c) lim f(x) (d) lim f(x) (e) f(5)
> A function is a ratio of quadratic functions and has a vertical asymptote x = 4 and just one x-intercept, x = 1. It is known that f has a removable discontinuity at x = -1 and limx→1 f (x) = 2. Evaluate (a). f (0) (b). limx→∞ f (x)
> The number of bacteria after t hours in a controlled laboratory experiment is n = f (t). (a). What is the meaning of the derivative f' (5)? What are its units? (b). Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do yo
> The cost of producing x ounces of gold from a new gold mine C = f (x) is dollars. (a). What is the meaning of the derivative f'(x)? What are its units? (b). What does the statement f' (800) = 17 mean? (c). Do you think the values of f' (x) will increase
> (a). Graph the function f (x) = ex + ln |x – 4| for 0 < x < 5. Do you think the graph is an accurate representation of ? (b). How would you get a graph that represents f better?
> (a). Prove that the equation has at least one real root. (b). Use your calculator to find an interval of length 0.01 that contains a root. cos x = x3
> Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. sin x = x2 - x (1, 2)
> The number N of US cellular phone subscribers (in millions) is shown in the table. (Midyear estimates are given.) (a). Find the average rate of cell phone growth (i). from 2002 to 2006 (ii). from 2002 to 2004 (iii). from 2000 to 2002 In each case, incl
> Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. 3√x = 1 – x, (1, 0)
> 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. x — х У- - 6х + 5 x? — 6х + 5
> Suppose f is continuous on [1, 5] and the only solutions of the equation f (x) = 6 are x = 1 and x = 4. If f (2) = 8, explain why f (3) > 6.
> (a). Find the slope of the tangent line to the curve y = x – x3 at the point (1, 0) (i). using Definition 1 (ii). using Equation 2 (b). Find an equation of the tangent line in part (a). (c). Graph the curve and the tangent line in successively smaller v
> If f (x) = x2 + 10 sin x, show that there is a number c such that f (c) = 1000.
> Each limit represents the derivative of some function f at some number a. State such an f and a in each case. 1* + t - 2 lim t - 1
> Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, find a function g that agrees with f for x ≠a and is continuous at a. x* – 1 (a) f(x) = a = 1 X - 1 x' - x? – 2x (b) f(x) =
> Find the values of and that make continuous everywhere. x? – 4 - 4 if x<2 x - 2 f(x) = ax? – bx + 3 if 2 <x<3 2х — а + b if x> 3
> For what value of the constant is the function continuous f on (-∞, ∞)? Scx? - сх? + 2х if х<2 f(x): x³ - cx if x> 2 — сх
> Use a graph to determine how close to 2 we have to take to ensure that x3 – 3x + 4 is within a distance 0.2 of the number 6. What if we insist that x3 - 3x + 4 be within 0.1 of 6?
> Find the numbers at which the function is discontinuous. At which of these points is continuous from the right, from the left, or neither? Sketch the graph of f. x + 2 f(x) = {e* х+2 if x<0 if 0 <xs1 2 — х if x>1
> (a). Use numerical and graphical evidence to guess the value of the limit limx→1 x3 – 1/√x - 1 (b). How close to 1 does have to be to ensure that the function in part (a) is within a distance 0.5 of its limit?
> Show that f is continuous on (-∞, ∞). x? if x<1 Vx if x> 1 f(x) =
> (a). Evaluate h (x) = (tan x-x)/x3 for x = 1, 0.5, 0.1, 0.05, 0.01, and 0.005. (b). Guess the value of limx→0 tan x – x/x3. (c). Evaluate h (x) for successively smaller values of until you finally reach a value 0 of for h (x). Are you still confident tha
> (a). Find the slope of the tangent line to the parabola y = 4x – x2 at the point (1, 3) (i). using Definition 1 (ii). using Equation 2 (b). Find an equation of the tangent line in part (a). (c). Graph the parabola and the tangent line. As a check on you
> Find f' (a). f (t) = 2t + 1/t + 3
> Locate the discontinuities of the function and illustrate by graphing. y = ln (tan2x)
> Use continuity to evaluate the limit. limx→4 5 + √x/√5 + x
> (a). If G (x) = 4x2 – x2, find G' (x) and use it to find equations of the tangent lines to the curve y = 4x2 – x3 at the points (2, 8) and (3, 9). (b). Illustrate part (a) by graphing the curve and the tangent lines on the same screen.
> The point P (1, ½) lies on the curve y = x (1 + x). (a). If Q is the point (x, x/ (1 + x), use your calculator to find the slope of the secant line PQ correct to six decimal places) for the following values of x: (i). 0.5 (ii). 0.9 (iii). 0.99 (iv). 0
> (a). Evaluate the function f (x) = x2 – (2x/100) for x = 1 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of limx→0 (x2 – 2x/1000) (b). Evaluate f (x) for x = 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again.
> The slope of the tangent line to the graph of the exponential function y = 2x at the point (0, 1) is limx→0 (2x – 1)/x. Estimate the slope to three decimal places.
> (a). Estimate the value of the limit limx→0 (1 + x)1/x to five decimal places. Does this number look familiar? (b). Illustrate part (a) by graphing the function y = (1 + x)1/x.
> (a). Estimate the value of limx→0 = sinx/sinπx by graphing the function f (x) = (sin x)/(sinπx). State your answer correct to two decimal places. (b). Check your answer in part (a) by evaluating f (x) for values of that approach 0.
> (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?
> 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 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.
> 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.
