2.99
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Rainbows are created when raindrops scatter sunlight. They have fascinated mankind since ancient times and have inspired attempts at scientific explanation since the time of Aristotle. In this project we use the ideas of Descartes and Newton to explain the shape, location, and colors of rainbows.

1. The figure shows a ray of sunlight entering a spherical raindrop at A. Some of the light is reflected, but the line AB shows the path of the part that enters the drop. Notice that the light is refracted toward the normal line AO and in fact Snellâ€™s Law says that sin a = k sin Î², where a is the angle of incidence Î², is the angle of refraction, and k â‰ˆ 4/5is the index of refraction for water. At B some of the light passes through the drop and is refracted into the air, but the line BC shows the part that is reflected. (The angle of incidence equals the angle of reflection.) When the ray reaches C, part of it is reflected, but for the time being we are more interested in the part that leaves the raindrop at C. (Notice that it is refracted away from the normal line.) The angle of deviation D (a) is the amount of clockwise rotation that the ray has undergone during this three-stage process. Thus

Show that the minimum value of the deviation is D (a) â‰ˆ1300 and occurs when a â‰ˆ59.40. The significance of the minimum deviation is that when a â‰ˆ59.40 we have D'(a) â‰ˆ 0, so âˆ†D/âˆ†a â‰ˆ 0. This means that many rays with a â‰ˆ 59.40 become deviated by approximately the same amount. It is the concentration of rays coming from near the direction of minimum deviation that creates the brightness of the primary rainbow. The following figure shows that the angle of elevation from the observer up to the highest point on the rainbow is 1800 â€“ 1380 = 420. (This angle is called the rainbow angle.)

2. Problem 1 explains the location of the primary rainbow, but how do we explain the colors? Sunlight comprises a range of wavelengths, from the red range through orange, yellow, green, blue, indigo, and violet. As Newton discovered in his prism experiments of 1666, the index of refraction is different for each color. (The effect is called dispersion.) For red light the refractive index is k â‰ˆ 1.3318 whereas for violet light it is k â‰ˆ 1.3435. By repeating the calculation of Problem 1 for these values of k, show that the rainbow angle is about 42.30 for the red bow 40.60 and for the violet bow. So the rainbow really consists of seven individual bows corresponding to the seven colors. 3. Perhaps you have seen a fainter secondary rainbow above the primary bow. That results from the part of a ray that enters a raindrop and is refracted at , reflected twice (at B and C), and refracted as it leaves the drop at D (see the figure at the left). This time the deviation angle D (a) is the total amount of counterclockwise rotation that the ray undergoes in this four-stage process. Show that

Taking k = 4/3, show that the minimum deviation is about 1290 and so the rainbow angle for the secondary rainbow is about 510, as shown in the following figure.

4. Show that the colors in the secondary rainbow appear in the opposite order from those in the primary rainbow.

A from B D(a --- sun or -- -- to observer Formation of the primary rainbow D(a) = (a – B) + (7 – 2B) + (a – B) = + 2a – 4B %3D rays from sun 138° rays from sun 42° --- -- observer D(a) = 2a – 6B + 27 and D(a) has a minimum value when k² – 1 cos a = 8 to --- observer from sun Formation of the secondary rainbow --- -- 42° 51°

** >** (a). Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2. (b). Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2. (c). Sketch the graph of a function that has a

** >** Let f (t) be the temperature at time where you live and suppose that at time t =3 you feel uncomfortably hot. How do you feel about the given data in each case?
(а) f'(3) — 2, f"(3) — 4 (Б) f (3) — 2, f"(3) — —4 (c) f'(3) = -2, f"(3) = 4 (d) f'(3) =

** >** If z2 = x2 + y2, dx/dt = 2, and dy/dt = 3, find dz/dt when x = 5 and y = 12.

** >** If x2 + y2 = 25 and dy/dt = 6, find dx/dt when y = 4.

** >** Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. Absolute minimum at 2, absolute maximum at 3, local minimum at 4

** >** C (x) = x1/3 (x + 4) (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check

** >** Suppose y = √2x + 1, where x and y are functions of t. (a). If dx/dt = 3, find dy/dt when x = 4. (b). If dy/dt = 5, find dx/dt when x = 12.

** >** The radius of a sphere is increasing at a rate of 5 mm/s. How fast is the volume increasing when the diameter is 80 mm?

** >** A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3/min. How fast is the height of the water increasing?

** >** A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station. (a). What quantities a

** >** Two people start from the same point. One walks east at 3 mi/h and the other walks northeast at 2 mi/h. How fast is the distance between the people changing after 15 minutes?

** >** A plane flying with a constant speed of 300 km/h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 300. At what rate is the distance from the plane to the radar station increasing a minute later?

** >** Find the absolute maximum and absolute minimum values of f on the given interval.
x? - 4 f(x) = [-4, 4] x² + 4'

** >** A Ferris wheel with a radius of 10m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level?

** >** (a). Find the intervals on which f is increasing or decreasing. (b). Find the local maximum and minimum values of f. (c). Find the intervals of concavity and the inflection points. f (x) = x4 - 2x2 + 3

** >** The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?

** >** A plane flies horizontally at an altitude of 5km and passes directly over a tracking telescope on the ground. When the angle of elevation is π/3, this angle is decreasing at a rate of π/6 rad/min. How fast is the plane traveling at that time?

** >** A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?

** >** B (x) = 3x2/3 - x (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check you

** >** Use the graph to state the absolute and local maximum and minimum values of the function.
y=g(x) 1

** >** A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into a

** >** Find the critical numbers of the function. f (x) = 4 + 1/3 x – 1/2 x2

** >** Brain weight B as a function of body weight W in fish has been modeled by the power function B = 0.007W2/3, where B and W are measured in grams. A model for body weight as a function of body length L (measured in centimeters) is W = 0.12L2.53. If, over 1

** >** Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.)
_f(x) = In x, 0<xs 2

** >** If two resistors with resistances R1 and R2 are connected in parallel, as in the figure, then the total resistance R, measured in ohms (â„¦), is given by 1/R = 1/R1 + 1/R2. If R1 and R2 are increasing at rates of 0.3â„¦

** >** When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV1.4 = C, where C is a constant. Suppose that at a certain instant the volume is 400 cm3 and the pressure is 80 kPa and is decreasin

** >** Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 600 cm3, the pressure is 150 kPa, and th

** >** Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 20/min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 600?

** >** Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is π/3.

** >** (a). Find the intervals on which f is increasing or decreasing. (b). Find the local maximum and minimum values of f. (c). Find the intervals of concavity and the inflection points. f (x) = 4x3 + 3x2 - 6x + 1

** >** A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?

** >** A (x) = x√x + 3 (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check your

** >** Each side of a square is increasing at a rate of 6m/s. At what rate is the area of the square increasing when the area of the square is 16 cm2?

** >** Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.)
(4 — х? if -2 <x<0 |2х — 1 if 0 <x <2 if -2 <х<0 f(x) =

** >** Find the absolute maximum and absolute minimum values of f on the given interval.
f(t) = t/4 – 1², [-1, 2]

** >** Use the graph to state the absolute and local maximum and minimum values of the function.
|y= f(x) 1t 1

** >** Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10

** >** A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. A cross-section is shown in the figure. If the pool is being filled at a rate of 0.8 ft3/min, how fast is the water level rising when the depth a

** >** A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft3 /min, how fast is the water level rising when the water is 6

** >** Water is leaking out of an inverted conical tank at a rate of 10,000 cm3 /min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate

** >** How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall?

** >** For what values of c is the function increasing on (-∞, ∞)? f (x) = cx + 1/x2 + 3

** >** (a). If the function f (x) = x3 + ax2 + bx has the local minimum value -2/9√3 at 1/√3, what are the values of a and b? (b). Which of the tangent lines to the curve in part (a) has the smallest slope?

** >** For what values of does the polynomial P (x) = x4 + cx3 + x2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as decreases?

** >** Find the critical numbers of the function. f (x) = x3 + x2 + x

** >** (a). Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b). Use a graph of f" to give better estimates. f (x) = x3(x – 2)4

** >** Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = 1 - √x

** >** (a). Find the intervals on which f is increasing or decreasing. (b). Find the local maximum and minimum values of f. (c). Find the intervals of concavity and the inflection points. f (x) = 2x3 + 3x2 - 36x

** >** Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three x-intercepts x1, x2 and x3, show that the x-coordinate of the inflection point is (x1 + x2 + x3)/3.

** >** Show that the curves y = e-x and y = -e-x touch the curve y = e-x sin x at its inflection points.

** >** The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2?

** >** At 2:00 PM a car’s speedometer reads 30 mi/h. At 2:10 PM it reads 50 mi/h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi/h2.

** >** Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider f (t) = g (t) – h (t), where and are the position functions of the two runners.]

** >** Suppose that 3 < f'(x) < 5 for all values of x. Show that 18 < f (8) – f (2) < 30.

** >** Suppose that f (0) = -3 and f'(x) < 5 for all values of x. The inequality gives a restriction on the rate of growth of f, which then imposes a restriction on the possible values of f. Use the Mean Value Theorem to determine how large f (4) can possibly b

** >** f (x) = 2 + 2x2 - x4 (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check

** >** (a). Show that ex > 1 + x for x > 0. (b). Deduce that ex > 1 + x + 1/2x2 for x > 0. (c). Use mathematical induction to prove that for x > 0 and any positive integer n,
e>1+x + +- n! + 2!

** >** f (x) = ln (x4 + 27) (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check

** >** h (x) = (x + 1)5 - 5x - 2 (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. C

** >** Between 00C and 300C, the volume V (in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula V = 999.87 - 0.06426T + 0.0085043T2 - 0.0000679T3 Find the temperature at which water has its maximum density.

** >** f (x) = x - 2 cos x, -2 < x < 0 (a). Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b). Use calculus to find the exact maximum and minimum values.

** >** The graph of the first derivative f' of a function f is shown. (a). On what intervals is f increasing? Explain. (b). At what values of x does f have a local maximum or minimum? Explain. (c). On what intervals is f concave upward or concave downward? Ex

** >** A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s. (a). At what rate is his distance from second base decreasing when he is halfway to first base? (b). At what rate is his distance

** >** f (x) = x√x - x2 (a). Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b). Use calculus to find the exact maximum and minimum values.

** >** The family of bell-shaped curves occurs in probability and statistics, where it is called the normal density function. The constant Âµ is called the mean and the positive constant Ïƒ is called the standard deviation. For simplicity,

** >** A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S (t) = Atp e-kt is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual de

** >** (a). Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b). Use a graph of f" to give better estimates.
f(x) = cos x + cos 2x, 0 <x< 27

** >** Coulombâ€™s Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles wit

** >** The figure shows a beam of length L embedded in concrete walls. If a constant load W is distributed evenly along its length, the beam takes the shape of the deflection curve where E and I are positive constants. (E is Youngâ€™s modulus

** >** For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.
y. 0 a b c drs x

** >** A formula for the derivative of a function f is given. How many critical numbers does f have?
f'(x) = 5e-01|=| sinx – 1

** >** Find the critical numbers of the function. f (x) = x-2 ln x

** >** Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?

** >** (a). Find the vertical and horizontal asymptotes. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts (a)&aci

** >** f (x) = 2 + 3x - x3 (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Check y

** >** f (x) = 2x3 - 3x2 - 12x (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Che

** >** Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.)
f(t) = cos t, -37/2 <t< 37/2

** >** (a). Find the critical numbers of f (x) = x4(x – 1)3. (b). What does the Second Derivative Test tell you about the behavior of f at these critical numbers? (c). What does the First Derivative Test tell you?

** >** Suppose f is a continuous function defined on a closed interval [a, b]. (a). What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for f? (b). What steps would you take to find those maximum and minimum values?

** >** Find the critical numbers of the function. h (t) = 3t - arcsin t

** >** Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = ex

** >** Suppose f" is continuous on (-∞, ∞). (a). If f'(2) = 0 and f"(2) = -5, what can you say about f? (b). If f'(6) = 0 and f"(6) = 0, what can you say about f?

** >** Show that the curve y = (1 + x)/ (1 + x2) has three points of inflection and they all lie on one straight line.

** >** Show that tan x > x for 0 < x < π/2. [ Show that f (x) = tan x - x is increasing on (0, π/2).]

** >** For what values of the numbers and does the function have the maximum value f (2) = 1?
f(x) = axe" ахе

** >** Find a cubic function f (x) = ax3 + bx2 + cx + d that has a local maximum value of 3 at = -2 and a local minimum value of 0 at x= 1.

** >** Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you prefer?
f(x) = x? + 4 -2

** >** In the theory of relativity, the energy of a particle is where m0 is the rest mass of the particle, is its wave length, and h is Planckâ€™s constant. Sketch the graph of E as a function of Î». What does the graph say about the

** >** A particle moves along the curve y = √1 + x2. As it reaches the point (2, 3), the y-coordinate is increasing at a rate of 4cm/s. How fast is the x-coordinate of the point changing at that instant?

** >** In the theory of relativity, the mass of a particle is where m0 is the rest mass of the particle m, is the mass when the particle moves with speed relative to the observer, and c is the speed of light. Sketch the graph of m as a function of v.
mo m

** >** Find dy/dx and d2y/dx2. For which values of is the parametric curve concave upward?
x = cos 2t, y = cos t, 0 <t< T

** >** Find dy/dx and d2y/dx2. For which values of is the parametric curve concave upward?
x=t - 12t, y = t² – 1 = 1? – 1

** >** Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.)
f(x) = x², 0<x< 2

** >** Find the critical numbers of the function. f (θ) = 2 cos θ + sin2 θ

** >** Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph f".
tan-x f(x) 1+ x' .3

** >** Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph f".
x* + x' + 1 f(x) Vx? + x + 1

** >** Suppose the derivative of a function f is f'(x) = (x + 1)2 (x – 3)5 (x – 6)4. On what interval is f increasing?

** >** (a). Find the vertical and horizontal asymptotes. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts (a)&aci

** >** g (x) = 200 + 8x3 + x4 (a). Find the intervals of increase or decrease. (b). Find the local maximum and minimum values. (c). Find the intervals of concavity and the inflection points. (d). Use the information from parts (a)–(c) to sketch the graph. Chec

** >** (a). Find the vertical and horizontal asymptotes. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts (a)&aci