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1, rv2 = k (r v1, v2) = k(rv) Deﬁnition of Scalar Multiplication Deﬁnition of Scalar Multiplication 11.8 Vectors 1019 The remaining properties are proved similarly and are left as exercises. Our next example demonstrates how Theorem 11.19 allows us to do the same kind of algebraic manipulations with vectors as we do wi...
, v2) v = v1, v2 in standard position. x 1020 Applications of Trigonometry Plotting a vector in standard position enables us to more easily quantify the concepts of magnitude and direction of the vector. We can convert the point (v1, v2) in rectangular coordinates to a pair (r, θ) in polar coordinates where r ≥ 0. The ...
know from Section 11.4 that (0, θ) is a polar representation for the origin for any angle θ. For this reason, ˆ0 is undeﬁned. The following theorem summarizes the important facts about the magnitude and direction of a vector. Theorem 11.20. Properties of Magnitude and Direction: Suppose v is a vector. v ≥ 0 and v = 0 ...
20. In words, the equation v = vˆv says that any given vector is the product of its magnitude and its direction – an important v is a result of concept to keep in mind when studying and using vectors. The equation ˆv = solving v = vˆv for ˆv by multiplying11 both sides of the equation by 1 v and using the properties of...
’s made sure there are examples and exercises which use them. 1022 Applications of Trigonometry y 5 4 3 2 1 v 60◦ θ = 120◦ −3 −2 −1 1 2 3 x 2. For v = 3, −3 √ 3, we get v = (3)2 + (−3 √ ﬁnd the θ we’re after by converting the point with rectangular coordinates (3, −3 form (r, θ) where r = v > 0. From Section 11.4, we h...
Hence, ˆw = 25 + 20. 25 = w along with w = 5, 1, −2 = 1√ 5, − 2√ 5 = The process exempliﬁed by number 1 in Example 11.8.4 above by which we take information about the magnitude and direction of a vector and ﬁnd the component form of a vector is called resolving a vector into its components. As an application of this p...
� speed of the plane is approximately 184 miles per hour. To ﬁnd the true bearing, we need to ﬁnd the angle θ which corresponds to the polar form (r, θ), r > 0, of the point (x, y) = (175 cos(50◦) + 35 cos(−30◦), 175 sin(50◦) + 35 sin(−30◦)). Since both of these coordinates are positive,14 we know θ is a Quadrant I ang...
zero vector by the factor v to produce a unit vector is called ‘normalizing the vector,’ and the resulting vector ˆv is called the ‘unit vector in the direction of v’. The terminal points of unit vectors, when plotted in standard position, lie on the Unit Circle. (You should take the time to show this.) As a result, we...
ı v1ˆı x v = v1, v2 = v1ˆı + v2ˆ. We conclude this section with a classic example which demonstrates how vectors are used to model forces. A ‘force’ is deﬁned as a ‘push’ or a ‘pull.’ The intensity of the push or pull is the magnitude of the force, and is measured in Netwons (N) in the SI system or pounds (lbs.) in th...
T1 2, √ T1 2 3 = For the second support, we note that the angle 30◦ is measured from the negative x-axis, so the angle needed to write T2 in component form is 150◦. Hence T2 = T2 cos (150◦), sin (150◦) √ = − T2 2 3, T2 2 The requirement w + T1 + T2 = 0 gives us this vector equation. 0, −50 +, T1 2 T1 2 − T1 2 T2 2 √ 3...
5 61, − 3 2 √ 3 9. v = 3ˆı + 4ˆ, w = −2ˆ 10. v = 1 2 (ˆı + ˆ), w = 1 2 (ˆı − ˆ) In Exercises 11 - 25, ﬁnd the component form of the vector v using the information given about its magnitude and direction. Give exact values. 11. v = 6; when drawn in standard position v lies in Quadrant I and makes a 60◦ angle with t...
and makes an angle measuring arctan 4 3 with the negative x-axis 25. v = 26; when drawn in standard position v lies in Quadrant IV and makes an angle measuring arctan 5 12 with the positive x-axis In Exercises 26 - 31, approximate the component form of the vector v using the information given about its magnitude and d...
3 53. A small boat leaves the dock at Camp DuNuthin and heads across the Nessie River at 17 miles per hour (that is, with respect to the water) at a bearing of S68◦W. The river is ﬂowing due east at 8 miles per hour. What is the boat’s true speed and heading? Round the speed to the nearest mile per hour and express the...
�E. What is the speed and bearing of the ocean current? Round the speed to the nearest mile per hour and express the heading as a bearing, rounded to the nearest tenth of a degree. 58. A 600 pound Sasquatch statue is suspended by two cables from a gymnasium ceiling. If each cable makes a 60◦ angle with the ceiling, ﬁnd...
move? If so, is it heading due west? 63. Let v = v1, v2 be any non-zero vector. Show that 1 v v has length 1. 64. We say that two non-zero vectors v and w are parallel if they have same or opposite directions. That is, v = 0 and w = 0 are parallel if either ˆv = ˆw or ˆv = − ˆw. Show that this means v = k w for some n...
ˆv = − 91 25, 312 25, vector 3. v + w = 0, 3, vector v + w = 3, scalar w − 2v = −6, 6, vector v + w = 3 √ 5, scalar v w − wv = −6 √ √ 5, 6 5, vector wˆv = 4, −2, vector 4. v + w = 8, 9, vector 5. 6. v + w = √ v w − wv = −14 v + w = √ 145, scalar √ 3, 3, vector √ 3, scalar v + w = 2 √ 29, vector 29, 6 v w − wv = 8 √ 3, ...
√ 2 2, vector, vector wˆv = 6 5, 8 5 w − 2v = −, scalar, vector wˆv = 1 2, 1 2, vector 11. v = 3, 3 √ 3 14. v = 0, 12 17. v = − 7 2, 0 √ 20. v = 6, −2 3 12. v = √ 3 2 2, 3 √ 15. v = −2 3, 2 √ 2 2 13. v = √ 3 3 3, 1 √ 16. v = − 3, 3 18. v = −5 √ √ 3 3, −5 19. v = 0, −6.25 21. v = 5, −5 22. v = 2, 4 23. v = −1, 3 24. v ...
� 45◦ √ 49. v = 17, θ ≈ 284.04◦ 50. v ≈ 145.48, θ ≈ 328.02◦ 51. v ≈ 1274.00, θ ≈ 40.75◦ 52. v ≈ 121.69, θ ≈ 159.66◦ 53. The boat’s true speed is about 10 miles per hour at a heading of S50.6◦W. 54. The HMS Sasquatch’s true speed is about 41 miles per hour at a heading of S26.8◦E. 55. She should maintain a speed of abou...
(3)(1) + (4)(−2) = −5. Note that the dot product takes two vectors and produces a scalar. For that reason, the quantity v· w is often called the scalar product of v and w. The dot product enjoys the following properties. Theorem 11.22. Properties of the Dot Product Commutative Property: For all vectors v and w, v · w ...
proof of k(v · w) = v · (k w) as an exercise. 11.9 The Dot Product and Projection 1035 For the last property, we note that if v = v1, v2, then v · v = v1, v2 · v1, v2 = v2 where the last equality comes courtesy of Deﬁnition 11.8. The following example puts Theorem 11.22 to good use. As in Example 11.8.3, we work out t...
, the machinations required to expand (v − w) · (v − w) for vectors v and w match those required to expand (v − w)(v − w) for real numbers v and w, and hence we get similar looking results. The identity veriﬁed in Example 11.9.1 plays a large role in the development of the geometric properties of the dot product, which...
, as seen below The Law of Cosines yields v − w2 = v2 + w2 − 2v w cos(θ). From Example 11.9.1, we know v − w2 = v2 − 2(v · w) + w2. Equating these two expressions for v − w2 gives v2 + w2 −2v w cos(θ) = v2 −2(v· w)+ w2 which reduces to −2v w cos(θ) = −2(v· w), or v · w = v w cos(θ), as required. An immediate consequenc...
vˆv) = w v v. In this case, k = w v > 0. 1037 √ 3)2 = 11.9 The Dot Product and Projection 1. We have v· w = 3, −3 √ 3·− √ 3, 1 = −3 √ 36 = 6 and w = (− 3)2 + 12 = √ √ √ √ 3−3 3 = −6 √ 4 = 2, θ = arccos 3. Since v = −6 12 = arccos √ 3 32 + (−3 √ − 3 2 = 5π 6. 2. For v = 2, 2 and w = 5, −5, we ﬁnd v · w = 2, 2 · 5, −5 = ...
�nition, = 0. Conversely, the angle between v and w is π v· w if v and w are nonzero vectors and v · w = 0, then Theorem 11.24 gives θ = arccos = 2. By Theorem 11.23, v · w = v w cos π 2 v w arccos to provide a diﬀerent proof about the relationship between the slopes of perpendicular lines.3 2, so v ⊥ w. We can use The...
m1m2. Hence, v1 · v2 = 0 if and only if 1 + m1m2 = 0, which is true if and only if m1m2 = −1, as required. While Theorem 11.25 certainly gives us some insight into what the dot product means geometrically, there is more to the story of the dot product. Consider the two nonzero vectors v and w drawn with a common initi...
θ) = − cos(θ), which means p = v cos(θ) = −v cos(θ). Rewriting this last equation in terms of v and w as before, we get p = −(v · ˆw). Putting this together with ˆp = − ˆw, we get p = pˆp = −(v · ˆw)(− ˆw) = (v · ˆw) ˆw in this case as well. 11.9 The Dot Product and Projection 1039 T v w θ O θ −−→ OR p = R If the angle...
time for an example. 1 w Example 11.9.4. Let v = 1, 8 and w = −1, 2. Find p = proj w(v), and plot v, w and p in standard position. Solution. We ﬁnd v · w = 1, 8 · −1, 2 = (−1) + 16 = 15 and w · w = −1, 2 · −1, 2 = 1 + 4 = 5. Hence, p = v· w 5 −1, 2 = −3, 6. We plot v, w and p below. w· w w = 15 4In this case, the poin...
27, we take p = proj w(v) and q = v − p. Then p is, by deﬁnition, a scalar multiple of w. Next, we compute q · w. 5See Exercise 64 in Section 11.8. 11.9 The Dot Product and Projection 1041 q · w = (v − p) · w Deﬁnition of q. = v · w − p · w Properties of Dot Product · w Since p = proj w(v). ( w · w) Properties of Dot P...
� P Q To ﬁnd the work W done in this scenario, we need to ﬁnd how much of the force F is in the −−→ P Q. This is precisely what the dot product F · P Q represents. Since direction of the motion −−→ the distance the object travels is P QP Q, −−→ −−→ W = ( F · P Q) P Q = F · ( P Q cos(θ), where θ is the angle between the...
50ˆı = 50 1, 0 = 50, 0. We get feet in the positive direction, the displacement vector is W = F · 3. Since force is measured in pounds and distance is measured in feet, we get W = 250 3 foot-pounds. Alternatively, we can use the formulation W = F −−→ P Q cos(θ) to get W = (10 pounds)(50 feet) cos (30◦) = 250 3, 5 · 50...
and w = −3ˆı + 4ˆ 18. v = 20 and w = 1 2, − √ 3 2 and 21. A force of 1500 pounds is required to tow a trailer. Find the work done towing the trailer along a ﬂat stretch of road 300 feet. Assume the force is applied in the direction of the motion. 22. Find the work done lifting a 10 pound book 3 feet straight up into ...
establish a Triangle Inequality for vectors. In this exercise, we prove that u + v ≤ u + v for all pairs of vectors u and v. (a) (Step 1) Show that u + v2 = u2 + 2u · v + v2. (b) (Step 2) Show that \|u · v\| ≤ uv. This is the celebrated Cauchy-Schwarz Inequality.6 (Hint: To show this inequality, start with the fact that...
q = −2, 1 3, 3 and w = − 3, −1 √ √ 6. v = −3 v · w = 6 θ = 60◦ proj w(v. v = 1, 17 and w = −1, 0 8. v = 3, 4 and w = 5, 12 v · w = −1 θ ≈ 93.37◦ proj w(v) = 1, 0 q = 0, 17 v · w = 63 θ ≈ 14.25◦ proj w(v) = 315 169, 756 q = 192 169, − 80 169 169 9. v = −4, −2 and w = 1, −5 10. v = −5, 6 and w = 4, −7 v · w = 6 θ ≈ 74.7...
= −3ˆı + 4ˆ v · w = 33 θ ≈ 59.49◦ proj w(v) = − 99 q = 224 25, 168 25 25, 132 25 17 = 75◦ and − 4 √ √ 3−1 4 3, proj w(v) = 1+ 4 q = √ 3 1− 4 √, 1+ 4 3 18 = 105◦ and − 4 6 proj w(v) = √ 3 √ q = 2+ 8 √ √ 2− 8 6 √ √ 2+ 8 6, √ 6 2 20. v = 1 and 19. v = √ 3 2, 1 2 √ and w = √ v · w = − θ = 165◦ 2 6+ 4 √ proj w(v) = √ q = ...
plane, as shown below. y 5 4 3 2 1 P (x, y) = (f (t), g(t)) Q O x 1 2 3 4 5 The curve C does not represent y as a function of x because it fails the Vertical Line Test and it does not represent x as a function of y because it fails the Horizontal Line Test. However, since the bug can be in only one place P (x, y) at a...
‘a’. As we shall see, there are inﬁnitely many diﬀerent parametric represen- tations for any given curve. 2Here, the bug reaches the point Q at two diﬀerent times. While this does not contradict our claim that f (t) and g(t) are functions of t, it shows that neither f nor g can be one-to-one. (Think about this before ...
described in Section 8.7 to eliminate the parameter t and get an equation involving just x and y. To do so, we choose to solve the equation y = 2t − 1 for t to get t = y+1 − 3 or, after some rearrangement, (y + 1)2 = 4(x + 3). Thinking back to Section 7.3, we see that the graph of this equation is a parabola with vert...
. for 0 < t < π x = sin(t) y = csc(t) x = 1 + 3 cos(t) y = 2 sin(t) for 0 ≤ t ≤ 3π 2 1. To get a feel for the curve described by the system x = t3, y = 2t2 we ﬁrst sketch the graphs of x = t3 and y = 2t2 over the interval [−1, 1]. We note that as t takes on values in the interval [−1, 1], x = t3 ranges between −1 and 1...
= 2( 3 below. x 1 −1 1 t −1 y 2 1 y 2 1 −1 1 t −1 1 x x = t3, −1 ≤ t ≤ 1 y = 2t2, −1 ≤ t ≤ 1 x = t3, y = 2t2, −1 ≤ t ≤ 1 4You should review Section 1.6.1 if you’ve forgotten what ‘increasing’, ‘decreasing’ and ‘relative minimum’ mean. 11.10 Parametric Equations 1051 9 3, 1 and for t = ln(3) we get 2 2. For the system ...
= e2 ln(x/2) = eln(x/2)2 4. Or, we could recognize that y = e−2t = e−t2 2 2, we get y = x = x2 4 this way as well. Either way, the graph of x = 2e−t, y = e−2t for t ≥ 0 is a portion of the parabola y = x2 4 which starts at the point (2, 1) and heads towards, but never reaches,6 (0, 0). = x 2, and since x = 2e−t means ...
) is now decreasing, and up, since y = csc(t) is now increasing. We pass back through 1 6 back to the points with small positive x-coordinates and large 2, 2 when t = 5π 2, 2 when t = π 2. Once t = π 2 5The reader is encouraged to review Sections 6.1 and 6.2 as needed. 6Note the open circle at the origin. See the solut...
moving to the left and upwards towards (1, 2). For π 2 ≤ t ≤ π, x is decreasing, as is y, so the motion is still right to left, but now is downwards from (1, 2) to (−2, 0). On the interval π, 3π, x begins to increase, while y continues to decrease. Hence, the motion becomes left 2 to right but continues downwards, con...
etrize x = g(y) as y runs through some interval I, let x = g(t) and y = t and let t run through I. To parametrize a directed line segment with initial point (x0, y0) and terminal point (x1, y1), let x = x0 + (x1 − x0)t and y = y0 + (y1 − y0)t for 0 ≤ t ≤ 1. To parametrize (x−h)2 a2 + (y−k)2 b2 = 1 where a, b > 0, let x...
(x), interchange x and y and solve for y. Doing so gives us the equation x = y5 + 2y + 1. While we could attempt to solve this equation for y, we don’t need to. We can parametrize x = f (y) = y5 + 2y + 1 by setting y = t so that x = t5 + 2t + 1. We know from our work in Section 3.1 that since f (x) = x5 + 2x + 1 is an...
t = 2 − x. Substituting this into y = −3 + 8t gives y = −3 + 8t = −3 + 8(2 − x), or y = −8x + 13. We know this is the graph of a line, so all we need to check is that it starts and stops at the correct points. When t = 0, x = 2 − t = 2, and when t = 1, x = 2 − t = 1. Plugging in x = 2 gives y = −8(2) + 13 = −3, for an...
and y = 2 + 3 sin(t) into the equation x2 + 2x + y2 − 4y = 4 and show that the latter is satisﬁed for all t such that 0 ≤ t < 2π. 9 = 1, we identify cos(t) = x+1 3 and sin(t) = y−2 (−1 + 3 cos(t))2 + 2(−1 + 3 cos(t)) + (2 + 3 sin(t))2 − 4(2 + 3 sin(t)) 1 − 6 cos(t) + 9 cos2(t) − 2 + 6 cos(t) + 4 + 12 sin(t) + 9 sin2(t...
ellipse, as required. 2. Our ﬁnal answer is {x = 2 cos(t), y = 3 sin(t) for π 4 + y2 9 = 1 gives 4 cos2(t) 2 ≤ t ≤ 3π + 9 sin2(t) 9 4 + y2 4 We note that the formulas given on page 1053 oﬀer only one of literally inﬁnitely many ways to parametrize the common curves listed there. At times, the formulas oﬀered there nee...
of t = −2. The problem here is that the parametrization we have starts 2 units ‘too soon’, so we need to introduce a ‘time delay’ of 2. Replacing every occurrence of t with (t − 2) gives x = −(t − 2), y = (t − 2)2 for −2 ≤ t − 2 ≤ 1. Simplifying yields x = 2 − t, y = t2 − 4t + 4 for 0 ≤ t ≤ 3. 1056 Applications of Tri...
) for 0 ≤ −t < 2π, which simpliﬁes9 to {x = cos(t), y = − sin(t) for −2π < t ≤ 0. This parametrization gives a clockwise orientation, but t = 0 still corresponds to the point (1, 0); the point (0, −1) is reached when t = − 3π 2. Our strategy is to ﬁrst get the parametrization to ‘start’ at the point (0, −1) and then sh...
in Example 11.10.4 number 3, we know that clockwise motion along the Unit Circle starting at the point (0, −1) can be modeled by the equations {x = − sin(θ), y = − cos(θ) for 0 ≤ θ < 2π. (We have renamed the parameter ‘θ’ to match the context of this problem.) To model this motion on a circle of radius r, all we need ...
the positive x-axis as described above. Graph your answer using a calculator. Solution. We have r = 3 which gives the equations {x = 3(t − sin(t)), y = 3(1 − cos(t)) for t ≥ 0. (Here we have returned to the convention of using t as the parameter.) Sketching the cycloid by hand is a wonderful exercise in Calculus, but ...
0 and 6 messes up the geometry of the cycloid. The reader is invited to use the Zoom Square feature on the graphing calculator to see what window gives a true geometric perspective of the three arches.) 14Again, see page 959 in Section 11.5. 11.10 Parametric Equations 1059 11.10.1 Exercises In Exercises 1 - 20, plot t...
t) y = t for − π 2 ≤ t ≤ π 2 In Exercises 21 - 24, plot the set of parametric equations with the help of a graphing utility. Be sure to indicate the orientation imparted on the curve by the parametrization. 21. 23. x = t3 − 3t y = t2 − 4 x = et + e−t y = et − e−t for − 2 ≤ t ≤ 2 for − 2 ≤ t ≤ 2 22. 24. x = 4 cos3(t) y ...
4y2 + 24y = 0, oriented clockwise (Shift the parameter so t = 0 corresponds to (0, 0).) 39. the triangle with vertices (0, 0), (3, 0), (0, 4), oriented counter-clockwise (Shift the parameter so t = 0 corresponds to (0, 0).) 40. Use parametric equations and a graphing utility to graph the inverse of f (x) = x3 + 3x − 4...
nd the parametric equations for the ﬂight of the hammer. (Here, use g = 32 ft. s2.) When will the hammer hit the ground? How far away will it hit the ground? Check your answer using a graphing utility. 44. Eliminate the parameter in the equations for projectile motion to show that the path of the projectile follows the...
∞). 49. Show that {x(t) = cosh(t), y(t) = sinh(t) parametrize the right half of the ‘unit’ hyperbola x2 − y2 = 1. (Hence the use of the adjective ‘hyperbolic.’) 50. Compare the deﬁnitions of cosh(t) and sinh(t) to the formulas for cos(t) and sin(t) given in Exercise 83f in Section 11.7. 51. Four other hyperbolic funct...
= t y = t3 y for − ∞ < t < ∞ 4 3 2 1 −1 −1 −2 −3 −4 1 x 9. x = cos(t) y = sin(t) y 1 for − π 2 ≤ t ≤ π 2 −1 x 1 −1 10. x = 3 cos(t) y = 3 sin(t) y for 0 ≤ t ≤ π 3 2 1 −3 −2 −1 1 2 3 x 11. x = −1 + 3 cos(t) y = 4 sin(t) for 0 ≤ t ≤ 2π 12. x = 3 cos(t) y = 2 sin(t) + 1 for π 2 ≤ t ≤ 2π y 4 3 2 1 1 2 x −4 −3 −2 −1 −1 −2 ...
2 −3 −4 4 3 2 1 −4 −3 −2 −1 −1 1 2 3 4 x −2 −3 −4 11.10 Parametric Equations 1067 23. x = et + e−t y = et − e−t for − 2 ≤ t ≤ 2 24. x = cos(3t) y = sin(4t) for 0 ≤ t ≤ 2π 1 1 x y 7 5 3 1 −1 −3 −5 −7 26. 28. 30. 32. 34. −1 x = 5t − 2 y = −1 − 3t x = t − 2 y = 4t − t2 x = t2 − 6t y = 3 − t for 0 ≤ t ≤ 1 for 0 ≤ t ≤ 4 for...
, 0 ≤ t ≤ 1 4t − 4, 1 ≤ t ≤ 2 12 − 4t, 2 ≤ t ≤ 3 1068 Applications of Trigonometry 40. The parametric equations for the inverse are x = t3 + 3t − 4 y = t for − ∞ < t < ∞ 41. r = 6 cos(2θ) translates to x = 6 cos(2θ) cos(θ) y = 6 cos(2θ) sin(θ) for 0 ≤ θ < 2π. 42. The parametric equations which describe the locations of...
, 622 alkalinity of a solution pH, 432 amplitude, 794, 881 angle acute, 694 between two vectors, 1035, 1036 central angle, 701 complementary, 696 coterminal, 698 decimal degrees, 695 deﬁnition, 693 degree, 694 DMS, 695 initial side, 698 measurement, 693 negative, 698 obtuse, 694 of declination, 761 of depression, 761 o...
ant determination of, 312 formal deﬁnition of, 311 slant (oblique), 311 vertical formal deﬁnition of, 304 intuitive deﬁnition of, 304 location of, 306 augmented matrix, 568 average angular velocity, 707 average cost, 346 average cost function, 82 average rate of change, 160 average velocity, 706 axis of symmetry, 191 b...
32 conjugate of a complex number deﬁnition of, 288 properties of, 289 Conjugate Pairs Theorem, 291 consistent system, 553 constant function as a horizontal line, 156 formal deﬁnition of, 101 intuitive deﬁnition of, 100 constant of proportionality, 350 constant term of a polynomial, 236 continuous, 241 continuously comp...
ition of, 1034 distributive property of, 1034 geometric interpretation, 1035 properties of, 1034 relation to orthogonality, 1037 relation to vector magnitude, 1034 work, 1042 Double Angle Identities, 776 earthquake Richter Scale, 431 eccentricity, 522, 981 eigenvalue, 626 eigenvector, 626 ellipse center, 516 deﬁnition ...
��erence, 76 diﬀerence quotient, 79 domain, 45 even, 95 exponential, 418 Fundamental Graphing Principle, 93 identity, 168 increasing, 100 independent variable of, 55 inverse deﬁnition of, 379 properties of, 379 solving for, 384 uniqueness of, 380 linear, 156 local (relative) maximum, 101 local (relative) minimum, 101 l...
oid, 542 identity function, 367 matrix, additive, 579 Index 1075 matrix, multiplicative, 585 statement which is always true, 549 imaginary axis, 991 imaginary part of a complex number, 991 imaginary unit, i, 287 implied domain of a function, 58 inconsistent system, 553 increasing function interval deﬁnition of, 3 notat...
554 two variables, 549 linear function, 156 local maximum formal deﬁnition of, 102 intuitive deﬁnition of, 101 local minimum formal deﬁnition of, 102 intuitive deﬁnition of, 101 logarithm algebraic properties of, 438 change of base formula, 442 common, 422 general, “base b”, 422 graphical properties of, 423 inverse pr...
of a complex number deﬁnition of, 991 properties of, 993 multiplicity eﬀect on the graph of a polynomial, 245, 249 of a zero, 244 natural base, 420 natural logarithm, 422 natural number deﬁnition of, 2 set of, 2 negative angle, 698 Newton’s Law of Cooling, 421, 474 Newton’s Law of Universal Gravitation, 351 oblique as...
239 leading coeﬃcient, 236 leading term, 236 variations in sign, 273 zero lower bound, 274 multiplicity, 244 upper bound, 274 positive angle, 698 Power Reduction Formulas, 778 power rule for absolute value, 173 for complex numbers, 997 for exponential functions, 437 for logarithms, 438 for radicals, 398 for the modulu...
equation, 654 reduced row echelon form, 570 reference angle, 721 Reference Angle Theorem for cosine and sine, 722 for the circular functions, 747 reﬂection of a function graph, 126 of a point, 10 regression coeﬃcient of determination, 226 correlation coeﬃcient, 226 least squares line, 225 quadratic, 228 total squared ...
795, 882 period, 881 phase, 881 phase shift, 795, 881 properties of, 881 vertical shift, 881 slant asymptote, 311 slant asymptote determination of, 312 formal deﬁnition of, 311 slope deﬁnition, 151 Index of a line, 151 rate of change, 154 slope-intercept form of a line, 155 smooth, 241 sound intensity level decibel, 4...
and Lower Bounds Theorem, 274 upper triangular matrix, 593 variable dependent, 55 independent, 55 variable cost, 159 variation constant of proportionality, 350 direct, 350 inverse, 350 joint, 350 variations in sign, 273 vector x-component, 1012 y-component, 1012 addition associative property, 1015 commutative property...
times and see how they differ (i.e., how they have changed). The two times of comparison are usually called the ﬁnal time and the initial time. We really need to be precise about this, so here is what we mean: ∆ quantity = (value of quantity at ﬁnal time) − (value of quantity at initial time) ∆ time = (ﬁnal time) − (i...
cases when we have a constant rate, we often want to ﬁnd the total amount of change in the quantity over a speciﬁc time period. The key principle in the background is this: Total Change in some Quantity = Rate Time × (1.2) It is important to mention that this formula only works when we have a constant rate, but that w...
the problem solving process. Example 1.3.1. How much time do you anticipate studying precalculus each week? Solution. One possible response is simply to say “a little” or “way too much!” You might not think these answers are the result of modeling, but they are. They are a consequence of modeling the total amount of s...
min/mile; what is her speed? (b) Which is faster: 100 mph or 150 ft/s? (c) Gina’s salary is 1 cent/second for a 40 hour work week. Tiare’s salary is $1400 for a 40 hour work week. Who has a higher salary? (d) Suppose it takes 180 credits to get a You accumubaccalaureate degree. late credit at the rate of one credit pe...
the cost of the magazine was $4.” A ﬁrst step would be these “pseudo-equations”: (Book cost) > $10 and (Magazine cost) = $4. (a) John’s salary is $56,000 a year and he pays no taxes. (b) John’s salary is at most $56,000 a year and he pays 15% of his salary in taxes. (c) John’s salary is at least $56,000 a year and he ...
liacci receives phone orders for pizza delivery at a constant rate: 18 orders in a typical 4 minute period. How many pies are sold in 4 hours? Assume Pagliacci starts taking orders at 5 : 00 pm and the proﬁt is a constant rate of $11 on 10 orders. When will phone order proﬁt exceed $1,000? Problem 1.10. Aleko’s Pizza h...
single day? (Note: There are 7.5 gallons in one cubic foot. Dumping into Lake Washington has stopped; now it goes into the Puget Sound.) ter 1, 2, 5 and 20 years. Would tax rates increase or decrease over time? Congress claims that this law would ultimately cut peoples’ tax rates by 75 %. Do you believe this claim? Pr...
17. (a) Solve for t: 3t −7 = 11+ t. (b) Solve for a: 1 + 1 a = 3. q (c) Solve for x: √x2 + a2 = 2a + x. (d) Solve for t: 1 − t > 4 − 2t. (e) Write as a single fraction: 2 x − 1 x + 1 Chapter 2 Imposing Coordinates You ﬁnd yourself visiting Spangle, WA and dinner time is approaching. A friend has recommended Tiff’s Dine...
axis and y-axis. Unless we say otherwise, we will always adopt the convention that the positive x-axis consists of those numbers to the right of the origin on 11 12 CHAPTER 2. IMPOSING COORDINATES the x-axis and the positive y-axis consists of those numbers above the origin on the y-axis. We have just described the xy-...
number y on the y-axis and a line ℓ∗ parallel to the y-axis passing through the number x on the x-axis. The two lines ℓ and ℓ∗ will intersect 6 6 2.2. THREE FEATURES OF A COORDINATE SYSTEM 13 in exactly one point in the plane, call it P. This procedure describes how to go from a given pair of real numbers to a point i...
0,0), Both pictures illustrate how the points lie on a parabola in the xy- coordinate system, but the aspect ratio has changed. The aspect ratio is deﬁned by this fraction: 5, 16 5, 4 5, 9 5, 1 5, 1 5, 4 5, 9 − 1 − 2 − 3 25 25 25 25 25 25 25 25,,,,,, 1 2 3 4 aspect ratio def= length of one unit on the vertical axis len...
589 73.8906 week 21 22 23 24 25 26 27 28 29 30 sales 81.6617 90.2501 99.7418 110.232 121.825 134.637 148.797 164.446 181.741 200.855 week 31 32 33 34 35 36 37 38 39 40 sales 221.98 245.325 271.126 299.641 331.155 365.982 404.473 447.012 494.024 545.982 week 41 42 43 44 45 46 47 48 49 50 sales 603.403 666.863 736.998 81...
xy-coordinate system is called imposing a coordinate system: There will often be many possible choices; it takes problem solving experience to develop intuition for a “natural” choice. This is a key step in all modeling problems. Example 2.3.1. Return to the tossed ball scenario on page 1. How do we decide where to dr...
-axis Cliff. y-axis Path of tossed ball. Cliff. x-axis (a) Origin at the top of the ledge. (b) Origin at the bottom of the ledge. Path of tossed ball. y-axis y-axis Path of tossed ball. Cliff. x-axis Cliff. x-axis (c) Origin at the landing point. (d) Origin at the launch point. Figure 2.7: Choices when imposing an xy-c...
this Chapter with a discussion of direction and distance in the plane. To set the stage, think about the following analogy: Example 2.4.1. You are in an airplane ﬂying from Denver to New York. How far will you ﬂy? To what extent will you travel north? To what extent will you travel east? Consider two points P = (x1,y1...
agorean Theorem to write: d2 = (∆x)2 + (∆y)2; that is, d = (∆x)2 + (∆y)2, p which tells us the distance d from P to Q. In other words, d is the distance we would ﬂy if we had ﬂown along that line segment connecting the two points. As an example, if P = (1, 1) and Q = (5, 4), then ∆x = 5 − 1 = 4, ∆y = 4 − 1 = 3 and d = ...
1 y2 P = (x1, y1) Beginning point. d Q = (x2, y2) Ending point. x2 x1 x-axis \|∆x\| Figure 2.10: A different direction. If your algebra is a little rusty, a very common mistake may crop up when you are using the distance formula. For example, 32 + 42? = √32 + √42? p √9 + 16 = 3 + 4 = 7. 5 Notice, you have an impossible s...
,320 sec and recalling that 1 mile = 5,280 feet, we arrive at d = 129,600√2 feet = 183,282 feet = 34.71 miles. For the second question, we specify the distance being 1 mile and want to ﬁnd when this occurs. The idea is to set d equal to 1 mile and solve for 2.4. DISTANCE 21 t. However, we need to be careful, since the ...
Suppose N is the beginning point and M is the ending point; recall Equation 2.2 and Figure 2.10 on Page 19. What is ∆x? What is ∆y? (f) If ∆x=0, what can you say about the relationship between the positions of the two points M and N? If ∆y=0, what can you say about the relationship between the positions of the two poi...
cars after 3 hours 47 minutes. 5 mi (c) When are the two cars 1 mile apart? kayak reaches shore here Problem 2.6. Allyson and Adrian have decided to connect their ankles with a bungee cord; one end is tied to each person’s ankle. The cord is 30 feet long, but can stretch up to 90 feet. They both start from the same lo...
An ant is located at the position (15,0) in the same coordinate system. Assume the location of the spider after t minutes is s(t) = (1 + 2t,2 + t) and the location of the ant after t minutes is a(t) = (15 − 2t,2t). (a) Sketch a picture of the situation, indicating the locations of the spider and ant at times t = 0,1,2...
are the planes 300 miles apart? Problem 2.11. Here is a list of some algebra problems with ”solutions.” Some of the solutions are correct and some are wrong. For each problem, determine: (i) if the answer is correct, (ii) if the steps are correct, (iii) identify any incorrect steps in the solution (noting that the ans...
axis Figure 3.1: A typical curve. Typically, the kind of condition we will give involves an equation in two variables (like x and y). We consider the three simplest situations in this chapter: horizontal lines, vertical lines and circles. 3.1 The Simplest Lines Undoubtedly, the simplest curves in the plane are the hori...
. x-axis Notice, the points (x, y) on the line m are EXACTLY the ones that lead to solutions of the equation x = 3; i.e., take any point on this line, plug the coordinates into the equation x = 3 and you get a true statement. Because the equation does not involve the variable y and only speciﬁes that x = 3, y can take ...
What is the left-hand side of Equation 3.1? To picture all points in the plane of distance r from the origin, fasten a pencil to one end of a non-elastic string (a string that will not stretch) of length r and tack the other end to the origin. Holding the string tight, the pencil point will locate a point of distance ...
, k) is precisely all of the solutions (x,y) of the equation (x − h)2 + (y − k)2 = r2; i.e., the circle is the graph of this equation. y-axis (x, y) r (h, k) x-axis We refer to the equation in the box as the standard form of the equation of a circle. From this equation you know both the center and radius of the circle ...
“to cut.” So, the intersection of two curves is the place where the curves “cut into” each other; in other words, where the two curves cross one another. If the pictures of two curves are given to us up front, we can often visually decide whether or not they intersect. This is one good reason for drawing a picture of ...
, 40) in the coordinate system; the radius of the disc will be r = 60 feet. We need to ﬁnd the points of intersection P, Q, R, and S of the circle with the x-axis and the y-axis. The equation for the circle with r = 60 and center (h, k) = (20, 40) is (x − 20)2 + (y − 40)2 = 3600. To ﬁnd the circular disc intersection w...
.3.1 gives us a general approach to ﬁnding the intersection points of circles with horizontal and vertical lines; this will be important in the exercises. 3.4 Summary • • • Every horizontal line has equation of the form y = c. Every vertical line has equation of the form x = c. Every circle has equation of the form (x ...

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