Test1-ProblemBank_Fa24.pdf

Test 1 Problem Bank Sections 2.1-2.3, 2.5-2.8, 3.1, 3.2 The Squeeze Theorem Suppose that g(x) ≤ f (x) ≤ h(x) for all x in some open interval containing c, except possibly at x = c itself. Suppose also that...

Test 1 Problem Bank Sections 2.1-2.3, 2.5-2.8, 3.1, 3.2 The Squeeze Theorem Suppose that g(x) ≤ f (x) ≤ h(x) for all x in some open interval containing c, except possibly at x = c itself. Suppose also that lim g(x) = lim h(x) = L x→c x→c Then lim f (x) = L x→c Definition and interpretation of the derivative. If we have a function f (x) and a number a, we define the derivative of f at a, written f ′ (a), to be f ′ (a) = or We often read f ′ (a) as “f prime of a.” Physically, we interpret f ′ (a) as the instantaneous rate of change of f at x = a. Graphically, we visualize f ′ (a) as the slope of the line tangent to the graph of f at x = a. Differentiation Rules to Remember! Civilization advances by extending the number of important operations which we can perform without thinking about them. – Alfred Whitehead, mathematician & philosopher d Derivative of a Constant: (c) = dx d n Power Rule: (x ) = dx d Constant Multiple Rule: If c is a constant, then [cf (x)] = dx d Sum Rule: [f (x) + g(x)] = dx d Product Rule: [f (x) · g(x)] = dx   d f (x) Quotient Rule: = dx g(x) d x Derivative of the Natural Exponential Function: (e ) = dx 1  1. Summarize (a) How can you visualize an average rate of change graphically? (b) How can you visualize an instantaneous rate of change graphically? (c) Which of the rates of change (average or instantaneous) are we able to compute at the moment? What is our current way of “computing” the other? (d) How do we compute average speed? How do we compute average velocity? 2. Summarize (a) What does it mean for a function f (x) to be continuous at a point x = a in its domain? (b) What are the different kinds of discontinuities and how do we describe them using limits? f (x) 3. Summarize: if you’re trying to evaluate lim , and you first find that lim g(x) = 0, what else do x→a g(x) x→a you need to look at? 4. Summarize How many vertical asymptotes can a function have? How many horizontal asymptotes can a function have? 5. Summarize (a) What are three different lenses we use to make sense of the derivative at a point f ′ (a)? (b) What is the connection between continuity and differentiability? 6. For a function f (x), summarize the relationships among features (positive/negative/zero, increas- ing/decreasing/constant, etc.) of f , f ′ , and f ′′. Draw some pictures to help you along the way! 7. In 2013, Diana Nyad became the first person to swim the 110-mile Florida Straits unaided. Other swimmers questioned whether her swim was truly unaided; one of the things they looked at was her speed along the swim.(1) In response, Nyad released GPS data about her 53-hour swim. Here’s a very small sample of the data: Time (hours) 0 5 7 15 31 32 40 Distance (miles) 0 6.67 8.90 18.72 54.85 58.72 82.88 Find Nyad’s average speed between 5 and 7 hours into her swim. Also find her average speed between 31 and 32 hours into her swim. Why might these numbers have raised suspicion? 8. A rocket is launched straight up into the air. The function f (t) = 10(t + 2)3 − 80 describes the rocket’s height in meters t seconds after liftoff. (1) See http://www.nytimes.com/2013/09/09/sports/questions-and-doubt-after-a-record-swim-from-cuba-to-florida. html 2  (a) Without using a calculator, sketch the graph of f (t) for t between 0 and 7. (b) Find the average velocity of the rocket between t = 1 and t = 3 (be sure to include units). How would you represent this velocity graphically on your graph from (a)? Suppose you’d now like to know the rocket’s instantaneous velocity 5 seconds after liftoff. That is, you’d like to know how fast the rocket is going right at the instant t = 5. (c) Would it work to find the rocket’s average velocity between t = 5 and t = 5? Why or why not? (d) How could you use average velocities to approximate the rocket’s instantaneous velocity at t = 5? Use your idea to give a good estimate of the instantaneous velocity at t = 5. (e) Based on your answer to (d), what do you think the rocket’s instantaneous velocity 5 seconds after liftoff is? How would you visualize this instantaneous velocity on the graph you drew in (a)? 9. Average speed vs. average velocity. A swimmer is swimming a 100 m long race, which is one lap in a 50 m long pool. Let s(t) be his distance from the starting position t seconds after the start of the race. (a) Which of the following is a more reasonable graph for s(t)? Why? s s 100 50 80 40 60 30 40 20 20 10 t t 10 20 30 40 50 10 20 30 40 50 (A) (B) (b) According to the graph you chose, what was the swimmer’s average speed for the race? Average velocity for the race? (c) What was the swimmer’s average speed over the first 20 seconds of the race? Average velocity? (d) What was the swimmer’s average speed over the last 50 m of the race? Average velocity? (e) What is the difference between velocity and speed? 10. The maximum swimming speed of salmon depends on the 30 25 water temperature. Suppose S(T ) is the maximum swimming 20 speed (in cm/s) of salmon when the water temperature is T 15 degrees Celsius; the graph of S is shown at right. What 10 does the average rate of change of S on the interval [15, 20] 5 represent? What are its units? Graphically, how would you 5 10 15 20 25 30 represent this average rate of change? 3 11. Sketch the graph of f (x) = 9 − 4(x − 2)2 , and put the following quantities in ascending order (from smallest to largest), with “