Question: Let P and Q be polynomials. Find /
Let P and Q be polynomials. Find 
if the degree of P is (a) less than the degree of Q and (b) greater than the degree of Q.
Transcribed Image Text:
P(x) lim Q(x)
> Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = 23 and g’(5) = 4.
> Determine the infinite limit. x, lim →3 (х — 3)5
> For the function f graphed in Exercise 18: a. Estimate the value of f’(50). b. Is f’(10) > f’(30)? c. Is f’(60) > f(80) – f(40)/80 - 40? Explain.
> The graph of a function f is shown. a. Find the average rate of change off on the interval [20, 60]. b. Identify an interval on which the average rate of change off is 0. c. Which interval gives a larger average rate of change, [40, 60] or [40, 70]? d
> For the function g whose graph is given, arrange the following numbers in increasing order and explain your reasoning: g'(-2) g'(0) g'(2) g'(4) y. y=g(x) 1 3 4 2.
> The displacement (in feet) of a particle moving in a straight line is given by s = 1/2t2 - 6t + 23, where t is measured in seconds. a. Find the average velocity over each time interval: i. [4, 8] ii. [6, 8] iii. [8, 10] iv. [8, 12] b. Find the ins
> 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.
> If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters) after t seconds is given by H = 10t - 1.86t2. a. Find the velocity of the rock after one second. b. Find the velocity of the rock when t = a. c. When will
> 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.
> Shown are graphs of the position functions of two runners, A and B, who run a 100-meter race and finish in a tie. a. Describe and compare how the runners run the race. b. At what time is the distance between the runners the greatest? c. At what time
> a. A particle starts by moving to the right along a horizontal line; the graph of its position function is shown in the figure. When is the particle moving to the right? Moving to the left? Standing still? b. Draw a graph of the velocity function.
> 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, 1/2). c. Graph the curve and both tangents on a common screen.
> Determine the infinite limit. 2 - x lim X- i (x – 1)?
> 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. lim f(x) = -o, lim f(x) = 5, lim f(x) = -5 X -00 X00
> For the function g whose graph is given, state the following. (a) lim g(x) (b) lim_g(x) X-00 (c) lim g(x) (d) lim g(x) x→2- (e) lim g(x) (f) The equations of the asymptotes x→2+ YA -1
> For the function f whose graph is given, state the following. e. The equations of the asymptotes (a) lim f(x) (b) lim f(x) X -0 (c) lim f(x) (d) lim f(x) yA 1 1
> 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.
> Explain in your own words the meaning of each of the following. (a) lim f(x) = 5 (b) lim f(x) = 3 X00 X -00
> a. Prove that And if these limits exist. b. Use part (a) and Exercise 65 to find lim f(x) = lim f(1/t) lim f(x) lim f(1/t) 1 lim x sin x→0+
> Formulate a precise definition of Then use your definition to prove that lim f(x) = X -00 lim (1 + x³) X -0 8.
> Use Definition 9 to prove that lim e* = 0.
> Determine the infinite limit. x + 1 lim x-5 x - 5
> Prove, using Definition 9, that lim x' = 0.
> Use Definition 8 to prove that 1 0. lim X-0 X
> a. How large do we have to take x so that 1/x2 b. Taking r = 2 in Theorem 5, we have the statement Prove this directly using Definition 7. 1 lim .2
> For the limit illustrate Definition 9 by finding a value of N that corresponds to M = 100. lim vx In x = 00 X00
> For the limit illustrate Definition 8 by finding values of N that correspond to ε = 0.1 and ε = 0.05. 1 — 3x lim = 3 -2 x² + 1 x- X -00
> For the limit illustrate Definition 7 by finding values of N that correspond to ε = 0.1 and ε = 0.05. 1 – 3x lim Vx2 + 1 -3
> Use a graph to find a number N such that 3x? + 1 if x> N then 1.5 < 0.05 | 2x2 + x + 1
> a. 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. b. Can you solve part (a) without using a graphing device?
> In Chapter 9 we will be able to show, under certain assumptions, that the velocity v(t) of a falling raindrop at time t is v(t) = v*(1 - e-gt/v*) Where g is the acceleration due to gravity and v* is the terminal velocity of the raindrop. a. Find limt →
> Determine the infinite limit. x + 1 lim X-5+ x - 5
> 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 C(t) = 30t/200 + t b. What happens to
> Find limx → ∞ f(x) if, for all x > 1, 10e* – 21 ) < I – x^ /x – 1 <f(x) 2e*
> 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,
> a. Use the Squeeze Theorem to evaluate b. Graph f(x) = (sin x)/x. How many times does the graph cross the asymptote? sin x lim
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = x+(x2 – 1)-(x + 2)
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = (3 - x)(1 + x)2(1 - x)4
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = x3(x + 2)2(x - 1)
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = x4 - x6
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = 2x3 - x4
> A function f 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) lim f(x)
> a. Estimate the value of by graphing the function f(x) = (sin πx)/(sin πx). State your answer correct to two decimal places. b. Check your answer in part (a) by evaluating f(x) for values of x that approach 0. sin x lim X0 si
> The point Ps2, 21d lies on the curve y = 1/(1 – x). a. If Q is the point (x, 1/(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. 1.5 ii. 1.9 iii. 1.99 iv. 1.999 v
> Find a formula for a function that has vertical asymptotes x = 1 and x = 3 and horizontal asymptote y = 1.
> Find a formula for a function f that satisfies the following conditions: lim f(x) = 0, lim f(x) = -, f(2) = 0, %3D lim f(x) = ∞, lim f(x) = -00
> Make a rough sketch of the curve y = xn (n an integer) for the following five cases: i. n = 0 ii. n > 0, n odd iii. n > 0, n even iv. n v. n Then use these sketches to find the following limits. (а) lim x^ X0+ (b) lim x" X0- (c) lim x" (d)
> 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 b. By calculating values of f(x), give numerical estimates of the limits in part (a). c. Cal
> Estimate the horizontal asymptote of the function f(x) = 3x3 + 500x2/x3 + 500x2 + 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?
> 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. y = 2ex/ ex - 5
> 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. y = x3 – x/x2 - 6x + 5
> 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. y = 1 + x4/x2 - x4
> 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. y = 2x2 + x – 1/x2 + x - 2
> a. By graphing the function f(x) = (cos 2x - 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 x that approach 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. y = 2x2 + 1/3x2 + 2x - 1
> 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. y = 5 + 4x/x + 3
> 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.
> a. Estimate the value of 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. lim (Vx² + x + 1 + x) X -0
> For f(x) = 2/x – 1/ln x find each of the following limits. e. Use the information from parts (a)–(d) to make a rough sketch of the graph of f. (a) lim f(x) (b) lim f(x) X→0+ (c) lim f(x) (d) lim f(x) x→1- x→1+
> a. For f(x) = x/ln x find each of the following limits. b. Use a table of values to estimate c. Use the information from parts (a) and (b) to make a rough sketch of the graph of f. (i) lim f(x) (ii) lim f(x) (iii) lim f(x) X→0+ I→I+ lim f(x).
> Find the limit or show that it does not exist. lim [In(2 + x) – In(1 + x)]
> Find the limit or show that it does not exist. lim [In(1 + x²) – In(1 + x)]
> Find the limit or show that it does not exist. lim tan (In x)
> Find the limit or show that it does not exist. cos x) -2x lim (e
> 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. lim x' Inx -2
> Find the limit or show that it does not exist. sin?x 2. lim .2 + 1
> Find the limit or show that it does not exist. 1 — е* lim x0 1 + 2e*
> Find the limit or show that it does not exist. e t - e lim x0 e 3x + e -3x — е
> Find the limit or show that it does not exist. lim arctan(e*)
> Find the limit or show that it does not exist. 1 + x* lim x* + 1 x-0
> Find the limit or show that it does not exist. lim (x² + 2x²) X-0
> Find the limit or show that it does not exist. lim (e* + 2 cos 3x)
> Find the limit or show that it does not exist. x* – 3x? + x lim x + 2
> Find the limit or show that it does not exist. lim Vx2 + 1
> Find the limit or show that it does not exist. lim (Vx2 + ax Vx² + bx
> 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. lim x*
> Find the limit or show that it does not exist. lim (V4x2 + 3x + 2x)
> Find the limit or show that it does not exist. lim (/9x? + x – 3x)
> Find the limit or show that it does not exist. x + 3x² lim x0 4x - 1 .2
> Find the limit or show that it does not exist. Vx + 3x? lim 4х — 1
> Find the limit or show that it does not exist. V1 + 4x6 lim .3 X -0 2 - x
> Find the limit or show that it does not exist. V1 + 4x6 lim .3 2 -
> Find the limit or show that it does not exist. x2 lim /x* + 1
> Find the limit or show that it does not exist. (2x² + 1)² lim (x – 1)(x² + x)
> Find the limit or show that it does not exist. lim a 2t3/2 + 3t – 5
> Find the limit or show that it does not exist. VE + t? lim 2t – t?
> 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. 5' – 1 lim 0 t
> Find the limit or show that it does not exist. 4x3 + 6х2 — 2 lim 2x3 — 4х + 5 X-00
> Find the limit or show that it does not exist. x - 2 lim x-0 x2 + 1
> Find the limit or show that it does not exist. 1 - x? .2 lim .3 x→* x' - x + 1
> Find the limit or show that it does not exist. Зх — 2 lim х 2х + 3x >00
> Evaluate the limit and justify each step by indicating the appropriate properties of limits. 9x³ + 8x — 4 lim 3 — 5х + x3
> Evaluate the limit and justify each step by indicating the appropriate properties of limits. 2x? – 7 lim 5x? + x – 3
> a. Use a graph of f(x) = (1 – 2/x)2 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.
> Guess the value of the limit 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. .2 lim X→* 2*
> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) lim f(x) = 2, f(0) = 0, f is even -00, x→3
> Sketch the graph of an example of a function f that satisfies all of the given conditions. f(0) = 3, lim f(x) = 4, x>0- lim f(x) = 2, x→0+ lim f(x) lim f(x) x→4- lim f(x) -00, -00, 00, X -00 X→4+ lim f(x) = 3
> 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. sin 30 S lim 00 tan 20
