2.1 The Tangent and Velocity Problems

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1 2.1 The Tangent and Velocity Problems Tangents What is a tangent? Tangent lines and Secant lines Estimating slopes from discrete data: Example: 1. A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. t (min) V (gal)

2 (a) If P is the point (15, 250) on the graph of V, find the slopes of the secant lines P Q when Q is the point on the graph with t = 10 and t = 20. (b) Estimate the slope of the tangent line at P by averaging the slopes of the two secant lines. (c) Use a graph of the function to estimate the slope of the tangent line at P (represents rate at which water is flowing from the tank after 15 min). Average Velocity vs. Instantaneous Velocity Average velocity= Instantaneous velocity at time t= Example: 8. The displacement (in cm) of a particle moving back and forth along a straight line is given by the equation of motion s = 2 sin πt + 3 cos πt, where t is measured in seconds. (a) Find the average velocity during each period. (i) [1,2] (ii) [1,1.1] (iii) [1,1.01] (iv) [1,1.001] (b) Estimate the instantaneous velocity of the particle when t = 1. 2

3 2.2 The Limit of a Function Definition The limit of f(x), as x approaches a, equals L written as lim = L if f(x) can be made arbitrarily close to L by taking x sufficiently close to a (on either side) but not equal to a. Question What is the difference between f(a) = L and lim f(x) = L? Estimating using a table Example: Create a table of values to estimate the limit. 19. lim x 1 x 6 1 x 10 1 What about lim x 0 sin π x? 3

4 Conclusion: Although using a table is one way of estimating limits, this will not always give the correct answer. So, we need to examine some other methods. One-Sided Limits lim f(x) = L if and only if lim f(x) = L and lim f(x) = L + Example: For the given graph of g state the value for each of the quantities, if it exists. If it doesn t exist, explain why. 5. (a) lim t 0 g(t) (b) lim g(t) t 0 + (c) lim g(t) t 0 (d) lim t 2 g(t) (e) lim g(t) t 2 + (f) lim g(t) t 2 (g) g(2) (h)lim t 4 g(t) Summary 4

5 2.3 Calculating Limits Using the Limit Laws There are 11 limit laws listed on pages in your book. Look these over to know what you can do with limits. Examples of limit laws in action: 1. Given that lim f(x) = 3, lim g(x) = 0, and lim h(x) = 8, find the limits that exist. (a) lim [f(x) + h(x)] (c) lim 3 h(x) (f) lim g(x) f(x) (g) lim f(x) g(x) 2f(x) (h) lim h(x) f(x) Sometimes you can just plug in the numbers, and go. Other times algebraic manipulation (factoring, rationalizing the numerator, etc.) is needed. Examples: (a) lim x 2 x 2 4 x 2 (b) lim x 0 x 3 8 x 2 (c) lim x 3 x 3 8 x 2 (d) lim x 2 x 3 8 x 2 5

6 (e) lim x 1 x x 1 When can and can t you evaluate a limit just by plugging in the relevant value of x? Examples: ( lim x 0 + x 1 ) x 37. (a) (i) lim x 2 +[[x]] (ii) lim [[x]] (iii) lim [[x]] x 2 x 2.4 (b) If n is an integer, evaluate (i) lim and (ii) x n +[[x]] lim x n [[x]]. (c) For what values of a does lim [[x]] exist? 6

7 The Squeeze Theorem: If f(x) g(x) h(x) when x is near a (except possibly at a) and lim f(x) = lim h(a) = L then lim g(x) L. 30. Prove that lim x 0 + xe sin(π/x) = 0. 7

8 2.4 Continuity Definitions: A function f is continuous at a number a if lim f(x) = f(a). If f is not continuous at a, we say f is discontinuous at a. Also, f is continuous on an interval if f is continuous at every number in the interval. This implies that three things need to happen for f to be continuous at a What happens if 1 of these 3 properties is not satisfied? Discontinuous= Types of discontinuities 1. Removable 2. Infinite 3. Jump Example: Find the discontinuities (if any). List the domain of each function. a. f(x) = x 1 x 2 3x + 2 b. g(x) = { x 1, x 1 x + 2, x > 1 8

9 NOTE: If f and g are continuous at a and c is a constant, then f + g, f g, cf, fg, f/g (if g(a) 0), and f g are also continuous at a. This tells us the following functions are continuous on their domains: Continuous from the left and right: Intermediate Value Theorem The Intermediate Value Theorem: Suppose that f is continuous on a closed interval [a, b] and let N be any number between f(a) and f(b), where f(a) f(b). Then there exists a number c in (a, b) such that f(c) = N. IVT is used in a variety of ways! 1. Graphing Calculators 2. Proving that roots exist for a function 3. Tons of other applications 9

10 Example: Show that there exists a number whose cube is one more than the number itself. 10

11 2.5 Limits Involving Infinity Definitions 1. lim f(x) = if the values of f(x) can be made as large as we please by taking x sufficiently close to a. Similar definition for. 2. Let f be defined on (a, ). Then lim x to L as we like by taking x sufficiently large. Similar for. Note: We also use lim x f(x) = L if values of f(x) can be made as close f(x) = to indicate the function f is getting large as x becomes large. Similar remarks can be made by replacing either (or both) s with s. Some Important Examples 1. arctan x: 2. tan x: 3. e x : 4. ln x: 5. 1 x : 11

12 Asymptotes What is a vertical asymptote? How do we find them? What is a horizontal asymptote? How do we find them? Calculating Limits Examples: Compute the limits of y = x + 5 x 2 25 as x ±, and as x ±5. Graph the function. 12

13 Compute lim x (x2 x 4 + 2). Graph y = 2x3 16 x 3 27 after finding the vertical and horizontal asymptotes. 13

14 2.6 Tangents, Velocities, and Other Rates of Change Some of this should look familiar, we talked about it in. Recall our definitions of tangent line and secant line. Locally Linear The tangent line to the curve y = f(x) at the point P (a, f(a)) is the line through P with slope f(x) f(a) m = lim x a provided that limit exists. Can also use an alternate definition: m = lim h 0 f(a + h) f(a) h Example: 6. (a) Find the slope of the tangent line to the curve y = x 3 at the point ( 1, 1) (b) Find an equation of the tangent line in part (a). 14

15 Applications For all rates of change (including velocity) the instantaneous rate of change is equal to the. 26. The number N of locations of a popular coffeehouse chain is given in the table. Year N (a) Find the average rate of growth from (i) 2000 to 2002 (ii) 2000 to 2001 (iii) 1999 to 2000 (b) Estimate the instantaneous rate of growth in 2000 by taking the average of two average rates of change. What are its units? (c) Estimate the instantaneous rate of growth in 2000 by measuring the slope of the tangent. 15

16 2.7 Derivatives Definition of Derivative at a point a: f f(a + h) f(a) (a) = lim, h 0 h if this limit exists. This can also be written as: f f(x) f(a) (a) = lim. x a Example: Find the derivative of f(t) = 1 + t at the number a. Equation of tangent line y f(a) = f (a)(x a) Example: If f(t) = 1 + t, find f (3) and use it to find an equation of the tangent line to f at the point (3, 2). 16

17 Rate of Change y instantaneous rate of change= lim x 0 x = lim f(x 2 ) f(x 1 ) x 2 x 1 x 2 x 1 SO...the derivative f (a) = inst. rate of change of y = f(x) with respect to x when x = a. When derivative is large, then. When derivative is small, then. Example: 25. A particle moves along a straight line w/ equation of motion s = f(t) = t 4.9t 2, where s is measured in meters and t in seconds. Find the velocity and speed when t = 5. Note 1: Velocity= Note 2: Speed= 30. The quantity (in pounds) of a coffee that is sold at a price of p dollars per pound is Q = f(p). (a) What is the meaning of the derivative f (8)? What are its units? (b)is f (8) positive or negative? Explain. 17

18 2.8 The Derivative as Function Creating f from f: 4. Let s look at some other notations for the derivative: Definition: A function f is differentiable at a if f (a) exists. The function f is differentiable on an open interval (a, b) if it is differentiable at every number in the interval. Theorem If f is differentiable at a, then f is continuous at a. When is a function not differentiable?

19 The converse of the above theorem is not true. Can you give me an example? Higher Derivatives We can also take the second derivative, third derivative, etc. of a function. In order to take a higher derivative,. Notation: Position Functions and Higher Derivatives: 39. The graph shows three functions. Determine which one is the position function of a car, the velocity function of the car, and the acceleration. 19

20 2.9 What does f say about f? Increasing/Decreasing If f (x) > 0, then f is. If f (x) < 0, then f is. Example: 11. (a) On what intervals is f increasing or decreasing? (b) At what values of x does f have a local maximum or minimum? Concavity If f (x) > 0, then f is. If f (x) < 0, then f is. Example: 11. (c) On what intervals is f concave upward or downward? 20

21 An inflection point is a point on the graph where the second derivative changes sign, i.e. where the concavity changes. (d) State the x-coordinates of the point(s) of inflection. (e) Assuming f(0) = 0, sketch a graph of f. Example: Given the following graph of f, sketch f and f. 21

22 Example: 19. Sketch the graph of a function that satisfies: f (x) > 0 if x < 2, f (x) < 0 if x > 2, f ( 2) = 0, lim x 2 f (x) =, and f (x) > 0 if x 2. An antiderivative of f is a function F such that F = f. If F is an antiderivative of f, so is. Example: 28. The graph of the velocity of a car is shown. Sketch the graph of the position function. 22