2.1 The idea of Limits F19 PDF

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2.1 The Idea of Limits Recall that the average velocity of an object which travels a distance d over a time period of duration t is vavg = dt . Question 1. How do you measure velocity? On his last vacation day Sparky D Dragon biked along the lake. He biked a total of 8 miles in 2 hours. What was his average velocity? Question 2. State the average rate of change for the following scenarios, being sure to include units. (a) It rained 4 inches over an 8 hour period.

(b) At 2 PM, the temperature was 82 degrees. At 5 PM the temperature was 76 degrees.

Question 3. Let’s say you travel 130 miles in 2 hours. What is your average rate of change (of distance per unit of time), or average velocity?

Is this how fast you drove throughout that entire time?

What do you think the difference between average velocity and instantaneous velocity is?

Recall that if you know the position of an object as a function of time f (t), the distance traveled from time t0 to time t1 can be calculated as f (t1 ) − f (t0 ). This means that average velocity can be expressed in these terms as vavg =

f ( t1 ) − f ( t0 ) . t1 − t0

Question 4. Sparky D Dragon, like other dragons, is afraid of eels. On his bike ride along the lake Sparky takes a break. While he is enjoying the view of the city an eel swims right up next to him startles him. He gets back on his bike and pedals as fast as he can to escape the eel. Suppose the distance between Sparky and the eel is given by the function f (t) = 2t2 + 10 measured in feet, and time in seconds. Find the average velocity in ft/s between: (a) t = 0 and t = 4 seconds

(b) t = 1 and t = 2 seconds

(c) t = 1 and t = 1 + h seconds

Definition: We say that a real number L is the limit of f (x) as x approaches a, written lim f (x) = L, if we can make the values of f (x) arbitrarily close to L by x→a

choosing x close enough to a (but not necessarily equal to a). Example 1. Let f (x) = x2 . We have the following table of values: x 2.1 2.01 2.001 f (x) 4.41 4.0401 4.004001 Notice that if the x-value has k zeroes after the decimal point, then the f (x)-value does as well. Since a large number of zeroes after the decimal point means that the number is very close to a whole number, we see that by making x arbitrarily close to 2, we get that f (x) is arbitrarily close to 4. Question 5. Evaluate the following limits: (a) lim (4 + 2x) x→0

(b) lim (4 + 2h) h→0...