Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued)
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1 Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued)
2 Prove this Result
3 How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever the limit doesn t exist For a function to have a derivative at a point, it must be defined at that point
4 How Can a Derivative Not Exist? Some situations where the derivative doesn t exist 1) At a cusp or corner
5 How Can a Derivative Not Exist? Some situations where the derivative doesn t exist 2) At a point on the graph that has a vertical tangent line
6 How Can a Derivative Not Exist? Some situations where the derivative doesn t exist 3) At a point where the function is not continuous
7 A Homework Problem Ex 6 (book sec. 2.1 #1, pg 82): 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. a) If P is the point (15, 250) on the graph of V, find the slopes of the secant lines PQ when Q is the point on the graph with t = 5, 10, 20, 25, and 30.
8 A Homework Problem Ex 6 (book sec. 2.1 #1, pg 82): 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. b) Estimate the slope of the tangent line at P by averaging the slopes of two secant lines.
9 A Homework Problem Ex 6 (book sec. 2.1 #1, pg 82): 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. c) Use a graph of the function to estimate the slope of the tangent line at P.
10 A Homework Problem Ex 6 (book sec. 2.1 #1, pg 82):
11 The Derivative as a Function Ex 7: If f x = x 2, find f 1, f 2, f 3, f 4, and f 5. This problem would take 5 separate calculations to complete. For each calculation, you would plug in the number to the formula for derivative and calculate a usually long limit. So we won t do the problem this way. Instead we ll use a trick.
12 The Derivative as a Function Ex 7: If f x = x 2, find f 1, f 2, f 3, f 4, and f 5. Trick: Don t plug in any number for a. Leave the a in the formula and calculate the limit. Your answer will have a in it. Then plug in the numbers. This way instead of having to go 5 separate calculations, you do 1 calculation then plug in all the numbers to it.
13 The Derivative as a Function Ex 7: If f x = x 2, find f 1, f 2, f 3, f 4, and f 5.
14 Definition of Derivative (Formula 3) Given a function f, the derivative of f is the function f defined by (3) f x = lim h 0 f x + h f(x) h for all values of x for which this limit exists.
15 Definition of Derivative (Formula 3) Other notations for the derivative Derivative as a function: f x dy dx df dx Derivative at a point: f a dy dx x=a df dx x=a
16 The Derivative as a Function Ex 8: Use the limit definition of the derivative to find f (x) if f x = 3x 1 x+5
17 Ex 9: The Derivative as a Function a) Use the limit definition of the derivative to find df f x = 1 4x. b) Find the equation of the tangent line to f x at x = 0. dx if
18 Sketching the Graph of the Derivative Ex 10 (Sec. 2.8 hw #4, pg. 168): The graph of f is given below. Sketch the graph of f
19 Sketching the Graph of the Derivative Ex 11 (Sec. 2.8 hw #3c, pg. 168): The graph of f is given below. Sketch the graph of f
20 Sketching the Graph of the Derivative Ex 12 (Sec. 2.8 hw #3b, pg. 168): The graph of f is given below. Sketch the graph of f
21 Chapter 3 Differentiation Rules
22 Sec 3.1: Derivatives of Polynomials and Exponential Functions
23 Recall: Definition of Derivative as a Function (formula 3) Given a function f, the derivative of f is another function f ' defined as (3) f ( x) lim h 0 f ( x h) f ( x) h for all values of x where this limit exists.
24 Goal of sec. 3.1, 3.2 and 3.4: To come up with a list of rules to be able to find the derivative of a function without having to do a lengthy limit calculation each time. Note: All of these derivative rules come from the limit definition of the derivative.
25 Derivative Rules (from sec. 3.1) In what follows, c and n stand for real numbers, and f and g stand for functions. Derivatives of specific functions c = 0 cx = c x n = nx n 1 e x = e x (power rule)
26 Derivative Rules (from sec. 3.1) In what follows, c and n stand for real numbers, and f and g stand for functions. Derivatives of combinations of functions f + g = f + g f g = f g cf = cf (sum rule) (difference rule) (constant multiple rule)
27 Some Quickies Ex 1: Find 7 = 5x = x 8 = e x = d dx x4 = d dx π2 = d dx ex = d ln (2) = dx
28 Derivative Rules (from sec. 3.1) Notes When finding a derivative, often times you will have to first rewrite the function so that it is in an easier form to take the derivative of When taking a derivative, take the entire derivative all at once. Don t take the derivative of only part of the function and not the rest because then you will end up writing something incorrect
29 Derivative Rules (from sec. 3.1) Ex 2: Find d dx x7 4x 5 + 7e x 5 x + 10 x x 3
30 Ex 3: Find Derivative Rules (from sec. 3.1) g (x) if g x = 3x7 3 x x 4 + 1
31 Derivative Rules (from sec. 3.1) Ex 4: If h x = x 3 5x e x, find a) h x b) h (3) x c) d4 h dx 4 d) The equation of the tangent line to h at x = 0.
32 Sec 3.2: The Product and Quotient Rules
33 Product and Quotient Rules The goal of section 3.2 is to add to the following list: f + g = f + g (sum rule) f g = f g cf = cf fg =? (difference rule) (constant multiple rule) (product rule) f g =? (quotient rule)
34 Product and Quotient Rules Product Rule fg = f g + g f
35 Product and Quotient Rules Quotient Rule f g = f g g f g 2
36 Product and Quotient Rules The goal of section 3.2 is to add to the following list: f + g = f + g (sum rule) f g = f g cf = cf fg = f g + g f (difference rule) (constant multiple rule) (product rule) f g = f g g f g 2 (quotient rule)
37 Product and Quotient Rules Ex 5: Find f (x) if f(x) = x 4 e x
38 Product and Quotient Rules dy Ex 6: Find if y = x2 + 5 dx x 7 3x
39 Product and Quotient Rules Ex 7: Find x 2 e x 2x x xe x
40 Sec 3.3: Derivatives of Trigonometric Functions
41 Derivatives of Trigonometric Functions sin (x ) = cos (x) cos (x ) = sin (x) Prove these
42 Derivatives of Trigonometric Functions Ex 8: Find d dx tan (x)
43 Derivatives of Trigonometric Functions Derivatives of specific functions (continued) sin (x) = cos (x) cos (x) = sin (x) tan (x) = sec 2 (x) sec (x) = sec(x)tan (x) csc (x) = csc x cot(x) cot (x) = csc 2 (x)
44 Derivatives of Trigonometric Functions Ex 9: Find f (x) if f x = sec x 5e x cos x + 3x5 + xtanx x + e x
45 Derivatives of Trigonometric Functions Pattern on the derivative of sine and cosine (sin x ) = cos (x) (cos x ) = sin (x) ( sin x ) = cos (x) ( cos x ) = sin (x) (sin x ) = cos (x) and so on
46 Derivatives of Trigonometric Functions Pattern on the derivative of sine and cosine so if f x = sinx, f (multiple of 4) x = sinx and also if g x = cosx, f (multiple of 4) x = cosx
47 Derivatives of Trigonometric Functions Ex 10: If f x = sin (x) and g x = cos (x), find a) f (37) (x) a) g (1927) (x)
48 Sec 3.4: The Chain Rule
49 The Chain Rule The chain rule is the rule for finding the derivative of a composition of functions. I.e. When one function s formula is plugged into another function s formula. f g (x) = f(g x ) = f (g x ) g (x)
50 The Chain Rule Ex 11: Find the derivative of each of the following functions a) f x = 5x 2 + cosx b) g x = e cosx c) h x = x 4 x 14 2x + 9 e d) i x = sin (cos x ) e) j x = sin x cos (x)
51 The Chain Rule Ex 11: Find the derivative of each of the following functions f) k x = sec x g) l x = cos 2 (x) h) m x = e 7x+3 i) n x = sin (9x) j) p x = sinxe x sec tanx x2 + 5x e x2
52 The Chain Rule Ex 12: Find the derivative of each of the following functions x + sinx a) f x = 3 b) g x = 7 x The point: When taking the derivative of a fraction where the top or bottom is just a number, don t waste your time using the quotient rule because it takes to long and you can calculate these derivatives faster.
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