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 | 1. What does the phrase “off on a tangent” mean? Relate the common meaning of the phrase to the image of a tangent to a circle (use the slingshot example, if that helps). In what way does the zoom image of the tangent promote the opposite view of the relationship between a curve and its tangent? |
| | 2. In general, the
instantaneous velocity of an object cannot be computed directly; the limit process is the only way to compute velocity at an instant. Given this, how does a car's speedometer compute velocity? (Hint: Look this up in a reference book or on the Internet. An important aspect of the car's ability to do this seemingly difficult task is that it performs analog calculations. For example, the pitch of a fly's buzz gives us an analog device for computing the speed of a fly's wings, since pitch is proportional to speed.) |
| | 3. Look in the news media (TV, newspaper, Internet) and find references to at least five different rates. We have defined a rate of change as the limit of the difference quotient of a
function. For your five examples, state as precisely as possible what the original
function is. Is the rate given quantitatively or qualitatively? If it is given quantitatively, is the rate given as a percentage or a number? In calculus, we usually compute rates (quantitatively) as numbers; is this in line with the standard usage? |
| | 4. Sketch the graph of a
function that is discontinuous at x = 1. Explain why there is no tangent line at x = 1.
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| In exercises 5-8, sketch in a plausible tangent line at the given point (Hint: Mentally zoom in on the point and use the zoom image of the tangent).
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| 5.
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at x = 
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| 6.
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at x = 1
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| 7.
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at x = 0
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| 8.
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at x = 1
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| In exercises 9 and 10, estimate the
slope of the tangent line to the curve at x = 1.
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| 9.
| 10.
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| In exercises 11 and 12, list the points A, B, C and
D in order of increasing slope of tangent line.
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| 11.
| 12.
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| In exercises 13-16, compute the
slope of the secant line between the points at (a) x = 1 and x = 2, (b)
x = 2 and x = 3, (c) x = 1.5 and x = 2, (d) x = 2 and
x = 2.5, (e) x = 1.9 and x = 2 and (f) x = 2 and
x = 2.1 and (g) estimate the slope of the tangent line at x = 2.
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| 13. f (x) = x3-x | 14.  |
| 15. f (x) = cos x2 | 16. f (x) = tan 2x
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| In exercises 17-20, use a CAS or graphing calculator.
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| 17. On one graph, sketch the secant lines in exercise 13, parts (a) - (d), and the tangent line in part (g). |
| 18. On one graph, sketch the secant lines in exercise 14, parts (a) - (d), and the tangent line in part (g). |
| 19. Animate the secant lines in exercise 13, parts (a), (c) and (e), converging to the tangent line in part (g). |
| 20. Animate the secant lines in exercise 13, parts (b), (d) and (f), converging to the tangent line in part (g). |
| In exercises 21-28, find the equation of the tangent line to
y = f (x) at x = a. Graph y =
f (x) and the tangent line to verify that you have the correct equation.
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| 21. f (x) = x2-2, a = 1 | 22. f (x) = x2-2, a = 0 |
| 23. f (x) = x2-3x, a = -2 | 24. f (x) = x3+x, a = 1
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| 25.  | 26.  |
| 27.  | 28. 
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| In exercises 29-34, use graphical and numerical evidence to determine whether or not the tangent line to
y = f (x) exists at x = a. If it does, estimate the
slope of the tangent; if not, explain why not.
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| 29. f (x) = | x-1| at a = 1 | 30.  at a = 1
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| 31.  at a = 0
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| 32.  at a = 1
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| 33.  at a = 0
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| 34.  at a = 0
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| In exercises 35-38, the
function represents the position of an object at time
t seconds. Find the average velocity between (a) t = 0 and
t = 2, (b) t = 1 and t = 2, (c) t = 1.9 and
t = 2 and (d) t = 1.99 and t = 2 and (e) estimate the
instantaneous velocity at
t = 2.
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| 35. f (t ) = 16t 2+10 | 36. f (t ) = 3t 3+t |
| 37.  | 38. f (t ) = 100sin (t /4)
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| In exercises 39-42, use the position
function
f (t ) to find the velocity at time t =
a.
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| 39. f (t ) = -16t 2+5, a = 1 | 40. f (t ) = -16t 2+5, a = 2 |
| 41.  | 42. 
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| 43. The graph below shows the elevation of a person on a hike up a mountain as a
function of time. When did the hiker reach the top? When was the hiker going the fastest on the way up? When was the hiker going the fastest on the way down? What do you think occurred at places where the graph is level? |
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| 44. The graph below shows the amount of water in a city water tank as a
function of time. When was the tank the fullest? the emptiest? When was the tank filling up at the fastest rate? When was the tank emptying at the fastest rate? What time of day do you think the level portion represents? |
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| 45. Suppose a hot cup of coffee is left in a room for 2 hours. Sketch a reasonable graph of what the temperature would look like as a
function of time. Then sketch a graph of what the rate of change of the temperature would look like. |
| 46. Sketch a graph representing the height of a bungee-jumper. Sketch the graph of the person's velocity (use + for upward velocity and - for downward velocity). |
| 47. In using a slingshot, a different type of velocity is important during the preliminary rotations. Angular velocity is defined by  , where  (t ) is the angle of rotation at time t. If the angle of a slingshot is (t ) = 0.4t 2 , what is the angular velocity after three rotations? [Hint: Which value of t (radians) corresponds to three rotations?] |
| 48. Find the angular velocity of the slingshot in exercise 47 after two rotations. Explain why the third rotation is helpful. |
| 49. You may have noticed in exercise 35 that there is a wrong way to get the right answer. For
quadratic functions (but definitely not most other functions), the average velocity between t = r and t = s equals the average of the velocities at t = r and t = s. To show this, assume that f (t ) = at2+bt+c is the
distance function. Show that the average velocity between t = r and t = s equals a(s+r)+b. Show that the velocity at t = r is 2ar+b and the velocity at t = s is 2as+b. Finally, show that 
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| 50. Find a cubic function [try f (t ) = t 3+ ] and numbers r and s such that the average velocity between t = r and t = s is different from the average of the velocities at t = r and t = s.
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| 51. In example 1.5, we analyzed a population  Graph y = f (t ). Argue that the rate of change of the population (the
slope of the tangent line) gets smaller as t gets larger. To illustrate this, compute the rate of change at t = 3. Compare your answer to the rate of change at t = 2, found in
Example 1.5. |
| 52. In exercise 51, compute the percentage change at t = 3 and compare to
Example 1.5. |
| 53. Show that  (Hint: Let h = x-a. ) |
| 54. Use the second limit in exercise 53 to recompute the
slope in exercises 21 and 23. Which limit do you prefer? |
| 55. A car speeding around a curve in the shape of y = x2 skids off at the point  . If the car continues in a straight path, will it hit a tree located at the point  ? |
| 56. For the car in exercise 55, show graphically that there is only one skid point on the curve y = x2 such that the tangent line passes through the point  . |
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| 57. Many optical illusions are caused by our brain's (unconscious) use of the tangent line in determining the positions of objects. Suppose you are in the desert 100 feet from a palm tree. You see a particular spot 28 feet up on the palm tree due to light reflecting from that spot to your eyes. Normally, it is a good approximation to say that the light follows a straight line (top path in the figure). |
Two paths of light from tree to person. |
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| However, when there is a large temperature difference in the air, light may follow nonlinear paths. If, as in the desert, the air near the ground is much hotter than the air higher up, light will bend as indicated by the bottom path in the figure. Our brains always interpret light coming in straight paths, so you would think the spot on the tree is at y = 10 because of the top path and also at some other y because of the bottom path. |
Two perceived locations of tree. |
If the bottom curve is y = 0.002x2-0.24x+28 , find an equation of the tangent line at x = 100 and show that it crosses the y - axis at y = -10. That is, you would “see” the spot at y = 10 and also at y = -10 , a perfect reflection.
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| How do reflections normally occur in nature? From water! You would perceive a tree and its reflection in a pool of water. This is the desert mirage! |
| | 58. You can use a VCR to estimate speed. Most VCRs play at 30 frames per second, so with a frame-by-frame advance you can estimate time as the number of frames divided by 30. If you know the distance covered, you can compute the average velocity by dividing distance by time. Try this to estimate how fast you can throw a ball, run 50 yards, hit a tennis ball, or whatever speed you find interesting. Some of the possible inaccuracies are explored in the following exercise. |
| | 59. What is the peak speed for a human being? It has beenestimated that Carl Lewis reached a peak speed of 28 mph while winning a gold medal in the 1992 Olympics. Suppose that we have the following data for a sprinter. Wewantto estimate peak speed. We could start by computing
 but this is the average speed
over the entire race, not the peak speed. Argue that we want to compute average speeds only using adjacent measurements (e.g., 40 and 50 meters, or 50 and 56 meters). Do this for all 11 adjacent pairs and find the largest speed (if you want to convert to mph, divide by 0.447). We will explore how accurate this estimate might be below. |
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| Meters | Seconds |
| 30 | 3.2 |
| 40 | 4.2 |
| 50 | 5.16666 |
| 56 | 5.76666 |
| 58 | 5.93333 |
| 60 | 6.1 |
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| Meters | Seconds |
| 62 | 6.26666 |
| 64 | 6.46666 |
| 70 | 7.06666 |
| 80 | 8.0 |
| 90 | 9.0 |
| 100 | 10.0 |
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| Notice that all times are essentially multiples of 1/30 since the data was obtained using the VCR technique in exercise 58. Given this, why is it suspicious that all the distances are whole numbers? To get an idea of how much this might affect your calculations, change some of the distances. For instance, if you change 60 (meters) to 59.8, how much do your average velocity calculations change? One possible way to identify where mistakes have been made is to look at the pattern of average velocities: does it seem reasonable? Would a sprinter speed up and slow down in such a pattern? In places where the pattern seems suspicious, try adjusting the distances and see if you can produce a more realistic pattern. Taking all this into account, try to quantify your error analysis: what is the highest (lowest) the peak speed could be? |
