Chapter 1
Use the graph of the function f to decide whether the value of the given quantity exists. (If an answer does not exist, enter DNE.) (a) f(−2) ____ (b) lim x→−2 f(x) _____ (c) f(0) _____ (d) lim x→0 f(x) ____ (e) f(2) ____ (f) lim x→2 f(x) ____ (g) f(4) ____ (h) lim x→4 f(x) ____
Answer: (a) DNE (b) DNE (c) f(0) (d) DNE (e) DNE (f) 0.4 (g) -1 (h) DNE
Consider the function f(x) = √x and the point P(4,2) on the graph f. (a) Graph f and the secant lines passing through the point P(4, 2) and Q(x, f(x)) for x-values of 2, 6, and 7. (b) Find the slope of each secant line. (Round your answers to three decimal places.) ____ (line passing through Q(2, f(x))) ____ (line passing through Q(6, f(x))) ____ (line passing through Q(7, f(x))) (c) Use the results of part (b) to estimate the slope of the tangent line to the graph of f at P(4,2). ____ Describe how to improve your approximation of the slope. A) Choose secant lines that are nearly vertical. B) Choose secant lines that are nearly horizontal. C) Define the secant lines with points farther away from P. D) Define the secant lines with points closer to P.
Answer: Graph: √2 = 1.414 ; (2,1.414) √6 = 2.449 ; (6, 2.449) √7 = 2.646 ; (7, 2.646) Slope Formula; m = rise/run = y2 - y1/ x2 - x1: m1 = 2 - √2 /4 - 2 = 0.293 m2 = 2 - √6/4 - 6 = 0.225 m3 = 2- √7/4 - 7 = 0.215 Add: 0.225 + 0.215/2 = 0.22 D) Define the secant lines with points closer to P.
Consider the figures below. Use the rectangles in each graph to approximate the area of the region bounded by y = 5/x, y = 0, x = 1, and x = 5. (Round your answers to three decimal places.) ____ figure (1) ____ figure (2) Describe how you could continue this process to obtain a more accurate approximation of the area. A) Continually decrease the number of rectangles. B) Continually increase the number of rectangles. C) Continually increase the height of all rectangles. D) Continually decrease the height of all rectangles.
Answer:
Use the graphs to identify the values of c for which lim x→c f(x) does not exist. (Enter your answers as a comma-separated list.) (a) c = ____ (b) c = ____
Answer: (a) c = -3 (b) c = -2,0
Find the limits. f(x) = 2x^2 − 3x + 39, g(x) = 3√x + 5 (a) lim x→4 f(x) = ____ (b) lim x→59 g(x) = ____ (c) lim x→4 g(f(x)) = ____
Answer: (a) lim x→4 2x^2 − 3x + 39 = 59 (b) lim x→59 3√x + 5 = 4 (c) lim x→4 3√2x^2 - 3x + 39) + 5 = 4
Consider the following information. lim x→c f(x) = 8 Use the information to evaluate the limits. (a) lim x→c 3√f(x) = 3√8 = (b) lim x→c f(x)/12 = 8/12 = (c) lim x→c [f(x)]^2 = [8]^2 = (d) lim x→c [f(x)]^2/3 = [8]^2/3 =
Answer: (a) lim x→c 3√f(x) = 3√8 = 2 (b) lim x→c f(x)/12 = 8/12 = 2/3 (c) lim x→c [f(x)]^2 = [8]^2 = 64 (d) lim x→c [f(x)]^2/3 = [8]^2/3 = 4
Complete the table. (Round your answers to five decimal places. Assume x is in terms of radian.) x→0 8 sin x/x x; -0.1, -0.01, -0.001, 0, 0.001, 0.01, 0.1 f(x); ____, ____, ____, 0, ____, ____, ____ Use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. (Round your answer to five decimal places.) lim x→0 8 sin x/x ≈ ____
Answer: 1) lim x→0 8 sin (-0.1)/-0.1 = 7.98667 2) lim x→0 8 sin (-0.01)/-0.01 = 7.99987 3) lim x→0 8 sin (-0.001)/-0.001 = 8 5) lim x→0 8 sin (0.001)/0.001 = 8 6) lim x→0 8 sin (0.01)/0.01 = 7.99987 7) lim x→0 8 sin (0.1)/0.1 = 7.98667 lim x→0 8 sin x/x ≈ 8
Create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. (Round your answers to four decimal places. If an answer does not exist, enter DNE.) lim x→0 sin 8x/x x; -0.1, -0.01, -0.001, 0.001, 0.01, 0.1 f(x); ____, ____, ____, ____, ____, ____ lim x→0 sin 8x/x ≈ ____
Answer: 1) lim x→0 sin(-0.8)/-0.1 = 7.1736 2) lim x→0 sin(-0.08)/-0.01 = 7.9915 3) lim x→0 sin(-0.008)/-0.001 = 7.9999 5) lim x→0 sin(0.008)/0.001 = 7.9999 6) lim x→0 sin(0.08)/0.01 = 7.9915 7) lim x→0 sin(0.8)/0.1 = 7.1736 lim x→0 sin 8x/x ≈ 8
Consider the following limit. x→0 √x + 6 - √6/x Complete the table below. (Round your answers to four decimal places.) x; -0.1, -0.01, -0.001, 0.001, 0.01, 0.1 f(x); ____, ____, ____, ____, ____, ____ Use the table to estimate the limit. Use a graphing utility to graph the function to confirm your result. (Round your answer to four decimal places.) ____
Answer: 1) lim x→0 √(-0.1)+ 6 - √6/-0.1 = 0.2050 2) lim x→0 √(-0.01) + 6 - √6/-0.01 = 0.2042 3) lim x→0 √(-0.001) + 6 - √6/-0.001 = 0.2041 5) lim x→0 √(0.001)+ 6 - √6/0.001 = 0.2041 6) lim x→0 √(0.01) + 6 - √6/0.01 = 0.2040 7) lim x→0 √(0.1) + 6 - √6/0.1 = 0.2033 0.2041
Consider the following problem. Find the distance traveled in 10 seconds by an object traveling at a constant velocity of 30 feet per second. Decide whether the problem can be solved using precalculus or whether calculus is required. A) The problem can be solved using precalculus. B) The problem requires calculus to be solved. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. ____ ft
Answer: A) The problem can be solved using precalculus. 30 x 10 = 300 ft
Consider the following problem. A bicyclist is riding on a path modeled by the function f(x) = 0.08x, where x and f(x) are measured in miles (see figure). Find the rate of change of elevation at x = 4. Decide whether the problem can be solved using precalculus, or whether calculus is required. A) The problem can be solved using precalculus. B) The problem requires calculus to be solved. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. ____
Answer: A) The problem can be solved using precalculus. f(0) = 0.08 x 0 f(0) = 0 f(4) = 0.08 x 4 f(4) = 0.32 0.32 - 0/4 - 0 = 0.32/4 = 0.08
Sketch the graph of f. f(x) = {sin x, x < 0 6 − 6 cos x, 0 ≤ x ≤ 𝜋 cos x, x > 𝜋 Identify the values of c for which lim x → c f(x) exists. A) The limit exists at all points on the graph except where c = 0 and c = 𝜋. B) The limit exists at all points on the graph. C) The limit exists at all points on the graph except where c = 𝜋. D) The limit exists at all points on the graph except where c = 12.
Answer: C) The limit exists at all points on the graph except where c = 𝜋.
Consider the following. lim x → 1 3/x - 1 Use the graph to find the limit (if it exists). (If an answer does not exist, enter DNE.) ____
Answer: DNE
Consider the function f(x) = -6x + x^2 and the point P(2, -8) on the graph of f. (a) Graph f and the secant lines passing through P(2, -8) and Q(x, f(x)) for x-values of 3, 2.5, 1.5. (b) Find the slope of each secant line. ____(line passing through Q(3, f(x))) ____(line passing through Q(2.5, f(x))) ____(line passing through Q(1.5, f(x))) (c) Use the results of part (b) to estimate the slope of the tangent line to the graph of f at P(2, -8). ____ Describe how to improve your approximation of the slope. A) Choose secant lines that are nearly vertical. B) Define the secant lines with points farther away from P. C) Choose secant lines that are nearly horizontal. D) Define the secant lines with points closer to P.
Answer: Graph: -6(3) + (3)^2 = -9 ; (3,-9) -6(2.5) + (2.5)^2 = -8.75 ; (2.5,-8.75) -6(1.5) + (1.5)^2 = -6.75 ; (1.5,-6.75) Slope Formula; m = rise/run = y2 - y1/ x2 - x1: m1 = -8 - -9/2 - 3 = -1 m2 = -8 - -8.75/2 - 2.5 = -1.5 m3 = -8 - -6.75/2 - 1.5 = -2.5 Add: -1.5 + -2.5/2 = -2 D) Define the secant lines with points closer to P.
Consider the function. f(x) = |x + 2| - |x - 2|/x Estimate the following by evaluating f at x-values near 0. (Round your answer to four decimal places.) lim x→0 |x + 2| - |x - 2|/x ____ Sketch the graph of f.
Answer: lim x→0 |x + 2| - |x - 2|/x = 2
Find the limit. (If an answer does not exist, enter DNE.) lim x→22 √x + 3 − 5/x − 22 ____
Answer: lim x→22 √x + 3 − 5/x − 22 = 0.1
Find the limit. lim x→3 (5x − 4) ____
Answer: lim x→3 (5x − 4) = 11
Find the limit (if it exists). (If an answer does not exist, enter DNE.) lim x→6^+ |x - 6|/x - 6 ____
Answer: lim x→6^+ |x - 6|/x - 6 = 1
Use the Squeeze Theorem to find the following limit when b − |x − a| ≤ f(x) ≤ b + |x − a|. lim x→a f(x) ____
Answer: lim x→a b − |x − a| = b lim x→a b + |x − a| = b lim x→a f(x) = b
Find the limit of the trigonometric function. (If an answer does not exist, enter DNE.) lim 𝜃→0 cos(5𝜃) tan(5𝜃)/𝜃 _____
Answer: lim 𝜃→0 cos(5𝜃) tan(5𝜃)/𝜃 = 5
Complete the table. (Round your answers to four decimal places. Assume x is in terms of radian.) lim x→0 5 cos(x) − 5/x x; -0.1, -0.01, -0.001, 0, 0.001, 0.01, 0.1 f(x); ____, ____, ____, ?, ____, ____, ____ Use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. (Round your answer to four decimal places.) lim x→0 5 cos(x) − 5/x ≈ ____
Answer: 1) lim x→0 5 cos(-0.1) - 5/-0.1 = 0.2498 2) lim x→0 5 cos(-0.01) - 5/-0.01 = 0.0250 3) lim x→0 5 cos(-0.001) - 5/-0.001 = 0.0025 4) N/A 5) lim x→0 5 cos(0.001) - 5/0.001 = -0.0025 6) lim x→0 5 cos(0.01) - 5/0.01 = -0.0250 7) lim x→0 5 cos(0.1) - 5/0.1 = -0.2498 lim x→0 5 cos(x) − 5/x ≈ 0
Find an equation of the line that satisfies the given conditions. Through (4, 9); parallel to the line passing through (5, 7) and (1, 3) ____
Answer: Slope Formula; m = rise/run = y2 - y1/ x2 - x1: 3 - 7/1 - 5 = 1 9 = 4 + b b = 5 y = x + 5
Discuss the continuity of the function. f(x) = x^2 − 4/x + 2 f is discontinuous at x = ____
f is discontinuous at x = -2
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