Section 2.1 Limits: Approached Numerically and Graphically

Transcription

1 Section 2.1 Limits: Approached Numerically and Graphically Foundation Concepts: Limit Left-hand limit Right-hand limit 1 = 1 = tiny big Practice: 1. What can we say about lim,. f(x)? a) If lim, 3 4 f(x)=7 and lim f(x)=7, what can we say about lim f(x)?, 35, 3 b) If lim, 6 4 f(x)=7 and lim f(x)=9, what can we say about lim f(x)?, 65, 6 c) If lim f(x) =, what can we say about lim f(x)?, 78, 78

2 2. Use the graph on the right to find the following limits: a. lim f(x) =, 7; 5 b. lim f(x) =, 7; 4 c. lim, = f(x) = 3. Use the graph below to find lim, >/6 f(x) 4. Find the limit, if it exists: lim, 6 (x 6 + 8x 2) Problems Example 1 Use the following graph of G to find each limit. If the limit does not exist, state that fact. a) lim, 78 5 G(x) b) lim, 78 4 G(x) c) lim, 78 G(x) d) lim, = 5 G(x) e) lim, = 4 G(x) f) lim, = G(x) g) lim, 8 5 G(x) h) lim, 8 4 G(x) i) lim, 8 G(x) j) lim, ; G(x)

3 Example 2 Given Find a) lim, 8 5 F(x) b) lim, 8 4 F(x) c) lim, 8 F(x) Example 3 Given Find a) lim, 6 5 F(x) b) lim, 6 4 F(x) c) lim, 6 F(x)

4 Section 2.2 Limits and Continuity Foundation Concepts: Properties of Limits L1 lim,. c = L2 lim,. x = L3 lim,. cf(x) = L4 lim,. [f(x)] I = K L5 lim Jf(x) =,. If lim,. f(x) = L and lim,. g(x) = M, then we have the following. L6 lim,. [f(x) ± g(x)] = L7 lim,. [f(x) g(x)] = Q(,) L8 lim =,. R(R) Continuous

5 Practice: 1. Find lim, 6 (x 6 2x + 5). 2. Find lim, 6 x + 1. Use the graph below to answer questions f( 1) = ; lim f(x) = ; f is/is not (circle one) continuous at x = 1, 78 because. 4. f(1) = ; lim, 8 f(x) = ; f is/is not (circle one) continuous at x = 1 because. 5. f(3) = ; lim, ; f(x) = ; f is/is not (circle one) continuous at x = 3 because. Note: In the last space, cite the reason why the function is or is not continuous using the aspects of the definition of continuous.

6 6. Find the limit, if it exists: lim,w 7X, ;,7;. (Hint: the initiation substitution of x = 3 yields the form 0/0. Look for a way to simplify the function algebraically or use a table and/or graph to determine the limit.) 7. Find the limit, if it exists: lim, 8Y x 6 9. Problems Example 1 Use the graph below to answer the following questions. a) Find g( 1). b) Find lim g(x), lim g(x), and lim g(x)., 785 4, 78, 78 c) Is g continuous at x = 1? Why or why not? d) Find g(3).

7 e) Find lim, ; 5 g(x), lim g(x), and lim g(x)., ; 4, ; f) Is g continuous at x = 3? Why or why not? Example 2 Find lim, 8.,78, W 78. Example 3 Find lim. (8]\)^78 \ = \. Example 4 Find lim \ =. _ `4a 7_` \.

8 Section 2.3 Average Rates of Change Foundation Concepts: Average rate of change Difference quotient Practice: 1. The average price of a ticket to a minor league baseball game can be approximated by p(x) = 0.02x x , where x is the number of years after 1990 and p(x) is in dollars. (i) Find p(6). (ii) Find p(13). (iii) Find p(13) p(6). (iv) Find g(8;)7g(h), and interpret this result. 8;7h 2. Find the simplified form of the difference quotient for f(x) = 3x 2x 6. Show your work.

9 Problems Example 1 For f(x) = 2 2x 6, a) Find a simplified form of the difference quotient Q(,]\)7Q(,) \ b) Use the simplified form to compute the difference quotient for x = 1 and i. h = 0.5 ii. h = 0.1 iii. h = 0.01 iv. h = Example 2 If a rock is thrown from the ground with an initial velocity of 80 feet per second, then the height of the rock, in feet, at t seconds can be modeled by s(t) = 16t t, 0 t 5. a) Find s(2) s(0). b) Find m(6)7m(=). What does this quantity represent? 67=

10 Example 3 The number of unbreakable sunglasses sold at a specialty store during a sales promotion can be modeled by g(x) = 2x x, 0 x 10, where x represents the number of weeks since the promotion began and g(x) represents the number of sunglasses sold. a) Find g(2) g(0). What does this quantity represent? b) Find R(6)7R(=). What does this quantity represent? 67= c) Find g(10) g(8). What does this quantity represent? d) Find R(8=)7R(n). What does this quantity represent? 8=7n e) Compare the answers obtained in parts (b) and (d). Which is greater? Why?

11 Section 2.4 Differentiation Using Limits of Difference Quotients Foundation Conceptual: A. Give the definition of the derivative of f(x) using a limit of the difference quotient B. What is the slope of the tangent line to the graph of f(x) at x = a? Differentiable vs Nondifferentiable

12 Practice: 1. For f(x) = x 6 + 2x 5, find f p (x) using the difference quotient. 2. Given that f(x) = 6 and, fp (x) = 6 W, find the equation of the tangent line at x = 2., 3. If a rock is dropped from the top of a 196 ft tall building, the height of the rock, in feet, at t seconds can be modeled by s(t) = t 6, 0 t 3.5. Find v(t). Problems Example 1 For f(x) = x 6 x, a) Find a simplified form of the difference quotient Q(,]\)7Q(,). \ b) Find f p frx+hs f(x) (x) by lim using the simplified form in (a). \ = h

13 c) Compute f p (a) for the given values of a: i. a = 2 ii. a = 0 iii. a = 0.5 iv. a = 1 Example 2 For f(x) = x 6 2x + 3, a) Find f p frx+hs f(x) (x) by lim. \ = h b) Compute f p (a) for the given values of a: i. a = 2 ii. a = 0 iii. a = 1 c) Find the equation of the tangent line at point ra, f(a)s for the given values of a in part (b). i. a = 2 ii. a = 0 iii. a = 1

14 d) Graph the function and draw tangent lines at these points to confirm that the slopes match the derivatives obtained in part (b). Tip: If you have a graphing calculator, now would be a great time to use it!

15 Section 2.5 Basic Derivative Formulas and Properties Foundation Conceptual: A. Write two different notations to express the derivative of function f(x). B. If y = f(x) = C, then y p = f p (x) = (Derivative of a Constant Function) C. If y = f(x) = x v, then y p = f p (x) = (Power Rule) D. If y = f(x) = e,, then y p = f p (x) = (Natural Exponential Function Rule) E. If F(x) = k f(x), then y F(x) = (Constant Multiple Property) y, F. If F(x) = g(x) ± h(x), then y F(x) = (Sum and Difference Properties) y, G. Higher Order Derivatives Practice: 1. Find the derivative of y = 5x 6 2.1x. 2. Given that f(x) = 5x 6 7x + 9, find f p (x).

16 3. Differentiate y = 6 x. 4. Given y = ;, dy, compute., 6 dx 5. Given s(t) = t 6, find s p (t) and s pp (t). Problems Example 1 For each of the following functions, find f p (x). 5 a) f(x) = 3 Jx 3 b) f(x) = 6, W c) f(x) = 6,

17 Example 2 For each of the following functions, find y. a) y = 3, } +3 x 3 ^, W b) y =,W ; + ;, W c) y = 5e, x ~ Example 3 a) y W Differentiate. y, W (8x6 6x + 4) b) y y, x} c) y y, (3e, ) Example 4 Find the points on the curve where the tangent line is horizontal. If such a point does not exist, state that fact. a) f(x) = 8 6 x6 1 b) f(x) = 2x ; + 3x 6 12x + 1 c) f(x) = 3x 1

18 Example 5 If a rock is thrown from the ground with an initial velocity of 80 feet per second, then the height of the rock, in feet, at t seconds can be modeled by s(t) = 16t t, 0 t 5. a) Find v(t) and a(t). b) Find v(3). What does this quantity represent? c) At t = 3 seconds, is the rock rising up or falling down? How do you know? d) At what time does the rock reach the highest height, and what is that height? e) What is the acceleration of the rock when it reaches the highest height?

19 Section 2.6 Differentiation Techniques: Product & Quotient Rules Foundation Conceptual: A. If F(x) = f(x) g(x), then F (x) = (Product Rule) B. If F(x) = Q(,), then y F(x) = (Quotient Rule) R(,) y, Practice: 1. Differentiate g(x) = (x )(x 7; + 5). 2. For the function f(x) = (3x 6 + 2)e,, a. Find f p (x). b. Find f p (0). c. Find an equation of the tangent line to the graph of the function at (0,2).

20 ,^ 3. Differentiate (x) =,78. Problems Example 1 For each of the following functions, find f p (x). a) f(x) = 3 x + n, (e, + x 6 + 4) b) f(x) = 6, ~` Example 2 For y = 3 xe, + 6 yƒ, find., y,

21 Example 3 Differentiate: yw y, W (x6 e, ) Example 4 A men s suit manufacturer finds that the cost, in dollars, of producing x suits is given by C(x) = x. Find the derivative of the average cost per item. Example 5 Find an equation of the tangent line to the graph of the function f(x) = a) x = 0 Y, W ]6 at b) x = 1

22 Section 2.7 Extended Power and Natural Exponential Rules Foundation Conceptual: A. If F(x) = [g(x)] v, then F (x) = (Extended Power Rule) B. If F(x) = e R(,), then F (x) = (Extended Natural Exponential Rule) Practice: Differentiate the following functions. 1. f(x) = (3 3x) 6;= 2. f(x) = 2 4x 3 + 3x 6 3. f(x) = 2e ;,W ]6,

23 4. f(x) = 8 (3, W 7},7;) Problems Example 1 For each of the following functions, indicate which rule(s) (Power Rule, Natural Exponential Rule, Sum and Difference, Product Rule, Quotient Rule, Extended Power Rule, Extended Natural Exponential Rule, etc.) should be used to find f p (x). (You do not need to compute f p (x).) a) f(x) = (2x ; + 5x 6 3) } b) f(x) = 5e =.=h c) f(x) = 3e Y,^73, d) f(x) = (x + 1) } (3x 2) 6 e) f(x) = 4x 3 e ;,W f) f(x) = x x f) f(x) = 3,]; },76 ; h) f(x) = } (6,]8)^ Example 2 For each of the following functions, find f p (x). (Tip: See above!) a) f(x) = x x 6 + 1

24 b) f(x) = 3,]; },76 ; c) f(x) = } (6,]8)^ Example 3 Differentiate: yw y, W re,w s

25 Section 2.8 Chain Rule and Derivatives of Log Functions Foundation Conceptual: A. If y = F(x) = frg(x)s, call y = f(u) with u = g(x), then y p = F p (x) = or yƒ = (Chain Rule) y, B. D. y ln x = C. y ln g(x) = y, y, y y, log. x = E. y y, a, = Practice: 1. For f(x) = 5x ; + 8 and g(x) = 2x, a. Find (f g)(x). b. Find (g f)(x). 2. Differentiate: a. y = ln(9x ; x 6 ) b. f(t) = ln(t ; + t) 3

26 3. Differentiate: a. y = 3 7, b. y = log 3 7x Problems Example 1 For each of the following, find yƒ, yœ yƒ and. yœ y, y, a) y = u ; and u = 3e, + 2 b) y = e Œ and u = 3x c) y = ln u and u = x Y + 1 Example 2 Find y p for y = 3 x ln(6x).

27 Example 3 Differentiate: yw y, W (x6 ln x) Example 4 at (1,0). Find an equation of the tangent line to the graph of the function f(x) = (3x + 2) ln x Example 5 In one city, 35% of all aluminum cans distributed will be recycled each year. A juice company distributes 110,000 cans. The number still in use after time t, in years, is given by N(t) = 110,000(0.35) Ž. Find N (t).