10Areview1

10Areview1 - 10A Midterm 1 Review Martha Yip Disclaimer:...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 10A Midterm 1 Review Martha Yip Disclaimer: there may be typos! Functions: to each input x, assigns a unique output f (x) Any vertical line intersects the graph of a function in at most one point. Domain: all possible inputs into f Range: all possible outputs of f , (after the domain is specified) Example: . . ....... ....... . 4. . .... ..... . .... ..... . . . ... . .. . ... .. . . . ... . . . . . . . ◦... ....• Domain = [1; 7] 3. . ... . . .. . . ..... .... . .. . .. . . . . . . . . Range = [0; 1] (2; 4] ◦ 2. . . . . . . . . . .. . • 1. ...• . . .. . . ..... . . .. . . ... . . . .. . ..... . . . . ................•............................................................................. .............................................................................................. ..... . . . . . . 1 3 6 7 ‘ 1 Functions: to each input x, assigns a unique output f (x) Any vertical line intersects the graph of a function in at most one point. Domain: all possible inputs into f Range: all possible outputs of f , (after the domain is specified) Example: . . ....... ....... . 4. . .... ..... . .... ..... . . . ... . .. . ... .. . . . ... . . . . . . . ◦... ....• Domain = [1, 7] 3. . ... . . .. . . ..... .... . .. . .. . . . . . . . . Range = [0, 1] ∪ (2, 4] ◦ 2. . . . . . . . . . .. . • 1. ...• . . .. . . ..... . . .. . . ... . . . .. . ..... . . . . ................•............................................................................. .............................................................................................. ..... . . . . . . 1 3 6 7 1-a Inverse Functions If y = f (x) is a function, then any vertical line intersects the graph in at most one point. If f (x) is a function with domain D and range R, then the inverse function of f (x), denoted by f −1(x) if it exists, is a function with domain R and range D. f (x) has an inverse if any horizontal line intersects the graph in at most one point. Procedure for finding the inverse function for y = f (x). (1) Switch y and x. (2) Solve for y. 2 Functions check your understanding 1. The graph of f −1 is a reflection of f in the line |x| . 2. True or False? x is an even function. |x| 3. True or False? x is an odd function. 4. Can a function be both odd and even? 5. True or False? f (x) = x2 has an inverse. 6. True or False? −1 (x) = −√x. f f (x) = x2 with domain (−∞, 0] has an inverse 3 Functions check your understanding 1. The graph of f −1 is a reflection of f in the line y = x. |x| 2. True or False? x is an even function. |x| 3. True or False? x is an odd function. 4. Can a function be both odd and even? 5. True or False? f (x) = x2 has an inverse. 6. True or False? −1 (x) = −√x. f f (x) = x2 with domain (−∞, 0] has an inverse 3-a Functions check your understanding 1. The graph of f −1 is a reflection of f in the line y = x. |x| 2. True or False? x is an even function. |x| 3. True or False? x is an odd function. 4. Can a function be both odd and even? 5. True or False? f (x) = x2 has an inverse. 6. True or False? −1 (x) = −√x. f f (x) = x2 with domain (−∞, 0] has an inverse 3-b Functions check your understanding 1. The graph of f −1 is a reflection of f in the line y = x. |x| 2. True or False? x is an even function. |x| 3. True or False? x is an odd function. 4. Can a function be both odd and even? 5. True or False? f (x) = x2 has an inverse. 6. True or False? −1 (x) = −√x. f f (x) = x2 with domain (−∞, 0] has an inverse 3-c Functions check your understanding 1. The graph of f −1 is a reflection of f in the line y = x. |x| 2. True or False? x is an even function. |x| 3. True or False? x is an odd function. 4. Can a function be both odd and even? If this was true, then −f (x) = f (−x) = f (x), so 2f (x) = 0, so f (x) must be the zero function. 5. True or False? f (x) = x2 has an inverse. f (x) = x2 with domain (−∞, 0] has an inverse 6. True or False? − 1 ( x) = − √ x. f 3-d Functions check your understanding 1. The graph of f −1 is a reflection of f in the line y = x. |x| 2. True or False? x is an even function. |x| 3. True or False? x is an odd function. 4. Can a function be both odd and even? If this was true, then −f (x) = f (−x) = f (x), so 2f (x) = 0, so f (x) must be the zero function. 5. True or False? f (x) = x2 has an inverse. f (x) = x2 with domain (−∞, 0] has an inverse 6. True or False? − 1 ( x) = − √ x. f 3-e Functions check your understanding 1. The graph of f −1 is a reflection of f in the line y = x. |x| 2. True or False? x is an even function. |x| 3. True or False? x is an odd function. 4. Can a function be both odd and even? If this was true, then −f (x) = f (−x) = f (x), so 2f (x) = 0, so f (x) must be the zero function. 5. True or False? f (x) = x2 has an inverse. f (x) = x2 with domain (−∞, 0] has an inverse 6. True or False? − 1 ( x) = − √ x. f 3-f Transformations of functions 1. Composition: f (g (x)) or g (f (x)) 2. Vertical shift: g (x) = f (x) + k 3. Horizontal shift: g (x) = f (x − k) 4. Vertical stretch/shrink: g (x) = kf (x) 5. Reflection in the x-axis: g (x) = −f (x) 6. Reflection in the y-axis: g (x) = f (−x) 4 f (x) = x2. Sketch g (x) = −f (x + 1) + 4. . . . . . . . . . . . . . . . . . . . . . . . . . . − . . . . . . . . . . . . . . . . . . . . . . . . . . . . − . . . . . . . . . . . . . . . . . . . . . .. . . .. . . − . . .. . .. .. . . . . .. .. . .. . . . .. − .. . .. . . .. .. . . . .. . .. . ... . .. ... . .. . ....................................................... . ..........|.............................|..................|........................................ ...................|......................... . . . . . . . | | | | . . . . . . . . . . − . . . . . . . . . . . − . . . . . . . . . . − . . . . . . . . . . . − . . . . 5 f (x) = x2. Sketch g (x) = −f (x+1) + 4. . . . . . . . . . . . . . . . . . . . . . . . . . . −. . . . . . . . . .. . . . . .. . . . .. . . . .. . . −. . .. . .. . . . .. . .. . . . .. .. .. . .. . − .. . .. .. .. .. . .. .. .. . .. . . . .. . − .. .. . .. .. ... .. . . ... ... . ... .. . . .................................|....................|.......... ..........|.............................|.......................................................... ...................|................ . . . . . . . . . | | | . . . . . . . . . . − . . . . . . . . . . . − . . . . . . . . . . − . . . . . . . . . . . − . . . . 5-a f (x) = x2. Sketch g (x) = −f (x + 1) + 4. . . . . . . − . . . . . . . . . . . − . . . . . . . . . . − . . . . . . . . . . . − . . . . . . . ..........|.................................................|.........|............................... ...................|..................|.....................................|..................... . . .. . .. . . .. . . . . . |. | . ... . . .. . . ... ... .. . .. . . .. . .. . . − . ... . ... . .. .. .. .. .. . .. . . .. . .. −. . .. . . .. . .. .. . .. . .. . . . .. .. − .. . . .. . . . . . . . . . . . . . . . . . . . . . − .. . . . . . . . . . . . . . . . . . . . 5-b f (x) = x2. Sketch g (x) = −f (x + 1)+4. . . . . ........ . ......... − . .. ... .. . . . .. . .. ... . . ... . . . . . . .. . .. .. . − .. .. . .. .. . .. .. . . . . .. . .. − .. . . .. . .. . .. .. . .. . .. . .. . . .. .. −. .. . . . . . . . . . . . . . . . ................................................... . ................................................................................ ..........|.............................|.......... ...................| . . .| .. . | | | | . . . . . . . . . . . . . . . . . . . . . . . . . . − . . . . . . . . . . . − . . . . . . . . . . − . . . . . . . . . . . − . . . . 5-c Transformations of functions check your understanding 1. True or False? f (g (x)) means apply f to x, then apply g . 2. Let g (x) = 2f (x) − 1. To get the graph of g , first do then do . , 6 Transformations of functions check your understanding 1. True or False? f (g (x)) means apply f to x, then apply g . 2. Let g (x) = 2f (x) − 1. To get the graph of g , first do then do . , 6-a Transformations of functions check your understanding 1. True or False? f (g (x)) means apply f to x, then apply g . 2. Let g (x) = 2f (x) − 1. To get the graph of g , first stretch vertically by a factor of 2, then shift down by 1 unit. 6-b Linear Functions Linear functions are of the form y = mx + b, where m is the slope, b is the y -intercept The graph of a linear function is a straight line. ..... ..... ..... ..... ..... . ..... . ...... ..... .... .. . •..... ∆y . ... . (0, b) . ...... . ..... m = ∆x . ..... . . ..... . ..... . . ..... . ..... . ..... . ..... . . ..... . .... . . .............................................................. ............................................................. .................... .................... . .... ... ... ... ..... ..... . . . . 7 Linear functions check your understanding 1. If the slope m is positive, the function is . 2. If the slope m is negative, the function is . 3. If the slope m = 0, the graph is a line. 4. A line through the points (p, q ) and (c, d) has slope 8 Linear functions check your understanding 1. If the slope m is positive, the function is increasing. 2. If the slope m is negative, the function is . 3. If the slope m = 0, the graph is a line. 4. A line through the points (p, q ) and (c, d) has slope 8-a Linear functions check your understanding 1. If the slope m is positive, the function is increasing. 2. If the slope m is negative, the function is decreasing. 3. If the slope m = 0, the graph is a line. 4. A line through the points (p, q ) and (c, d) has slope 8-b Linear functions check your understanding 1. If the slope m is positive, the function is increasing. 2. If the slope m is negative, the function is decreasing. 3. If the slope m = 0, the graph is a horizontal line. 4. A line through the points (p, q ) and (c, d) has slope 8-c Linear functions check your understanding 1. If the slope m is positive, the function is increasing. 2. If the slope m is negative, the function is decreasing. 3. If the slope m = 0, the graph is a horizontal line. d−q . c−p 4. A line through the points (p, q ) and (c, d) has slope 8-d Exponential Functions Exponential functions are of the form P = P0at, where P0 is the initial quantity, if a > 1, have exponential growth if 0 < a < 1, have exponential decay Can also express the growth factor a as a = 1 + r, and write P = P0(1 + r)t, where r is the growth rate. if r > 0, have exponential growth if r < 0, have exponential decay 9 Exponential Functions Continuous rate of growth Keep in mind the example of compound interest: k nt B = P0 1 + → P0ekt n If k > 0, have exponential growth. If k < 0, have exponential decay. as n → ∞ 10 Half life, doubling time Half life is the time it takes for the exponentially decaying quantity to be reduced to half. 1 Given P = P0at, find t so that P = 2 P0. So, solve for t in 1 P0 = P0at 2 or 1 = at. 2 Doubling time: Given P = P0at, find t so that P = 2P0, so solve 2 = at. 11 Exponential functions check your understanding 1. Consider P = 25(0.6)t. The growth factor a is factor r is . The function is exponentially . The growth -ing. 2. Which grows faster? ex, or 10x? 3. Which grows faster? 1000(2x) or 3x? 12 Exponential functions check your understanding 1. Consider P = 25(0.6)t. The growth factor a is 0.6. The growth factor r is a − 1 = −0.4. The function is exponentially decaying. 2. Which grows faster? ex, or 10x? 3. Which grows faster? 1000(2x) or 3x? 12-a Exponential functions check your understanding 1. Consider P = 25(0.6)t. The growth factor a is 0.6. The growth factor r is a − 1 = −0.4. The function is exponentially decaying. 2. Which grows faster? ex, or 10x? 3. Which grows faster? 1000(2x) or 3x? 12-b Exponential functions check your understanding 1. Consider P = 25(0.6)t. The growth factor a is 0.6. The growth factor r is a − 1 = −0.4. The function is exponentially decaying. 2. Which grows faster? ex, or 10x? 3. Which grows faster? 1000(2x) or 3x? 12-c Linear vs Exponential functions: charts Linear, slope m x -2 -1 0 1 2 f(x) Exp, growth factor a x -2 -1 m f(x) | ♣ m ~ ¡ 1 a ♥ |+ |+2 |+3 0 1 2 m m ~¡ ~¡ ~¡ a a 2 3 a 13 Linear vs Exponential functions: charts Linear, slope m x -2 -1 0 1 2 f(x) ♣−m ♣ ♣+m ♣ + 2m ♣ + 3m Exp, growth factor a x -2 -1 0 1 2 f(x) ♥ · a−1 ♥ ♥·a ♥ · a2 ♥ · a3 13-a Logarithmic functions The inverse of the exponential function ax is the logarithm loga(x): eln x = x, ln(ex) = x. Properties ea+b = eaeb a−b = ea e eb ln(ab) = ln(a) + ln(b) ln( a ) = ln(a) − ln(b) b ln(xa) = a ln(x) ln(1) = 0 eab = (ea)b e0 = 1 14 sin θ, cos θ, tan θ defined on the unit circle . . . . . . ............... ................ ..... . . ..... . .... .... . . .. ... . (cos θ, sin θ) ...•. ... . .... ... . . .. . . .. ... . .. . .. .. . . .. . .. . .. . . .. . . . .. . .. . ... . . .. . . . .. . .. . . . . . . .. . . ... .. . . . ... . . .θ . . .. .. ... . . . . ... . . . ............................................................ . .......................................................... ........ ....... . .. . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . .. . .. .. . .. . . . . .. .. . .. . ... . . ... . . .. .... ... . .... ...... . ........ ...... . ....... . . ............. ............ . . . . . . . . . tan θ = sin θ cos θ A circle has 2π radians = 360 degrees. One radian is the angle whose arc has length equal to the radius. ( 360 ∼ 57 degrees) 2π 15 Trig functions, cont’d Sine or Cosine functions are of the form y = A sin (Bx) + C, where A is the amplitude (half the distance between max and min) and |2π| is the period (time it takes to complete one cycle) B C is the vertical shift and the graph oscillates about that value y = A cos (Bx) + C Identities: sin2(x) + cos2(x) = 1 cos x = sin (x + π ) 2 (Pythagoras on the unit circle) (horizontal shift of graphs) 16 Trig functions check your understanding 1. True or False? sin θ is an odd function. 2. True or False? cos θ is an odd function. 3. The range of sin θ is . 4. The domain of tan θ is . 5. The range of 7 sin(2θ) + 1 is . 17 Trig functions check your understanding 1. True or False? sin θ is an odd function. 2. True or False? cos θ is an odd function. 3. The range of sin θ is . 4. The domain of tan θ is . 5. The range of 7 sin(2θ) + 1 is . 17-a Trig functions check your understanding 1. True or False? sin θ is an odd function. 2. True or False? cos θ is an odd function. 3. The range of sin θ is . 4. The domain of tan θ is . 5. The range of 7 sin(2θ) + 1 is . 17-b Trig functions check your understanding 1. True or False? sin θ is an odd function. 2. True or False? cos θ is an odd function. 3. The range of sin θ is [−1, 1]. 4. The domain of tan θ is . 5. The range of 7 sin(2θ) + 1 is . 17-c Trig functions check your understanding 1. True or False? sin θ is an odd function. 2. True or False? cos θ is an odd function. 3. The range of sin θ is [−1, 1]. 4. The domain of tan θ is x = . . . , − π , π , 3π , 5π , . . .. 22 2 2 5. The range of 7 sin(2θ) + 1 is . 17-d Trig functions check your understanding 1. True or False? sin θ is an odd function. 2. True or False? cos θ is an odd function. 3. The range of sin θ is [−1, 1]. 4. The domain of tan θ is x = . . . , − π , π , 3π , 5π , . . .. 22 2 2 5. The range of 7 sin(2θ) + 1 is [−6, 8]. 17-e Polynomial functions Polynomial functions of degree n are of the form p(x) = anxn + an−1xn−1 + · · · + a1x + a0. If n is even, the graph of p(x) is even. p(x) has at most n zeros, and may not have any zeros at all. If n is odd, the graph of p(x) is odd. p(x) has at most n zeros, and has at least 1 zero. If r is a zero of p(x), then p(x) has the factor (x − r): p(x) = (x − r)(an n − 1 degree polynomial). 18 Rational functions Rational functions are of the form f ( x) = p ( x) , q (x) where both p(x) and q (x) are polynomials. To find the zeros of f (x): find the zeros of p(x). To find where f (x) is not defined: find the zeros of q (x). Note: f (x) has a vertical asymptote x = r if r is a zero of q (x) (assuming it isn’t also a zero of p(x)) To find any horizontal asymptotes, check the behaviour of f (x) as x → ∞ or −∞. Remember, the terms with the highest degrees dominate. 19 Rational functions Let −2(x + 1)(x − 3)(x − 7) −2x3 + 18x2 − 22x − 42 = f ( x) = . 3 − 6x2 − 15x + 100 2 x (x + 4)(x − 5) The zeros are x = −1, 3, 7. The vertical asymptotes are x = −4, 5. As x → ±∞, −2x3 = −2, f (x) behaves like 3 x so the horizontal asymptote is y = −2. 20 Polynomial and rational functions check your understanding 1. As x → −∞, x13 goes to 1 2. As x → −∞, x2 goes to . . 3. Does x2 have a horizontal asymptote? 21 Polynomial and rational functions check your understanding 1. As x → −∞, x13 goes to −∞. 1 2. As x → −∞, x2 goes to . 3. Does x2 have a horizontal asymptote? 21-a Polynomial and rational functions check your understanding 1. As x → −∞, x13 goes to −∞. 1 2. As x → −∞, x2 goes to 0. 3. Does x2 have a horizontal asymptote? 21-b Polynomial and rational functions check your understanding 1. As x → −∞, x13 goes to −∞. 1 2. As x → −∞, x2 goes to 0. 3. Does x2 have a horizontal asymptote? No. 21-c Limits If as x approaches c, the values of f (x) approach the number L, then L is the limit of the function f (x) as x approaches c. Notation: x→c lim f (x) = L. This is understood to be a two-sided limit. The left-hand and right-hand limits are x→c− lim f (x) and x→c+ lim f (x) where we only look at approaching c from the left or the right. The two-sided limit exists if and only if the one-sided limits exist and are equal. 22 Limits at infinity The concept of limits at infinity is the same as “end behaviour” and horizontal asymptotes. If as x approaches ∞, f (x) approaches a number L, then x→∞ x2 −5 Example: f (x) = 3x2−1x lim f (x) = L. 3x2 − 5x lim x→∞ x2 − 1 3x2 − 5x 3−5 2 2 x = lim x = lim 3 − 0 = 3. = lim x 2 x→∞ x x→∞ 1 − 1 x→∞ 1 − 0 1 − x2 2 2 x x 23 Continuity of a function f (x) is continuous at x = c if x→c lim f (x) = f (c). Example: Consider the function x2 − 4 (x − 2)(x + 2) f ( x) = = . x+2 x+2 Its domain is (−∞, −2) ∪ (−2, ∞). Away from x = −2, its graph is the straightline y = x − 2. f (x) is continuous everywhere except at x = −2. In order to make this function continuous everywhere, we should define f (−2) = (−2 − 2) = −4. Intuitively, the graph is ”unbroken” at x = c. 24 Limits laws If lim f (x) and lim g (x) each exist, then • Sums: lim f (x) ± g (x) = lim f (x) ± lim g (x) • Mul. by constant: lim kf (x) = k lim f (x). • Product Law: lim f (x)g (x) = lim f (x) lim g (x). • Quotient Law: if lim g (x) = 0, then lim lim f (x) f ( x) = . g ( x) lim g (x) 25 Limits and Continuity f ( x) . . . . . . . . . . . . . ... . . . . . ... . . . . . ... . . . ◦. 3 −− . . . . . . ... . . . . . . ... . . . . . ... . . • . 2 −− . . ... . . . . . . . ... . . . . . ... . . . . ... . . ... .. .. .. .. .. .. .. .. . .......................................................... ......................................................... . . | . . . . . . 3 Examples to keep in mind g ( x) . . . . . . . . . . . . . ... . . . . . ... . . . . . ... . . . •. 3 −− ... . . . . . ... . . . . . . ... . . . . . ... . . . . . ... . . . . . ... . . . . . ... . . . . . ... . . .... .. .. .. .. .. .. .. .. . .............................................................. .............................................................. . . | . . . . . . 3 As long as the function is continuous at c, then we can “plug in c” to find the limit. 26 Limits check your understanding 1. lim f (x) = L means that f (x) gets near to x→c ciently close to (but is different from) . when x gets suffi- 2. If lim f (x) = −2 and lim f (x) = −2, then lim f (x) = x→c+ x→c− x→c . 3. If lim f (x) = 1 and lim g (x) = 2, then lim 2f (x) + g (x) = x→2 x→2 x→2 . 4. 1 = x→2+ x − 2 lim . 27 Limits check your understanding 1. lim f (x) = L means that f (x) gets near to L when x gets sufficiently x→c close to (but is different from) c. 2. If lim f (x) = −2 and lim f (x) = −2, then lim f (x) = x→c+ x→c− x→c . 3. If lim f (x) = 1 and lim g (x) = 2, then lim 2f (x) + g (x) = x→2 x→2 x→2 . 4. 1 = x→2+ x − 2 lim . 27-a Limits check your understanding 1. lim f (x) = L means that f (x) gets near to L when x gets sufficiently x→c close to (but is different from) c. 2. If lim f (x) = −2 and lim f (x) = −2, then lim f (x) = −2. x→c+ x→c− x→c 3. If lim f (x) = 1 and lim g (x) = 2, then lim 2f (x) + g (x) = x→2 x→2 x→2 . 4. 1 = x→2+ x − 2 lim . 27-b Limits check your understanding 1. lim f (x) = L means that f (x) gets near to L when x gets sufficiently x→c close to (but is different from) c. 2. If lim f (x) = −2 and lim f (x) = −2, then lim f (x) = −2. x→c+ x→c− x→c 3. If lim f (x) = 1 and lim g (x) = 2, then lim 2f (x) + g (x) = 4. x→2 x→2 x→2 4. 1 = x→2+ x − 2 lim . 27-c Limits check your understanding 1. lim f (x) = L means that f (x) gets near to L when x gets sufficiently x→c close to (but is different from) c. 2. If lim f (x) = −2 and lim f (x) = −2, then lim f (x) = −2. x→c+ x→c− x→c 3. If lim f (x) = 1 and lim g (x) = 2, then lim 2f (x) + g (x) = 4. x→2 x→2 x→2 4. 1 = +∞, the limit does not exist. x→2+ x − 2 lim 27-d Example . . ..... ........ . 4. . .... ..... . .... ..... . . . . . .. ... . ... .. . . . ... . . . . . . ◦... ....• 3. ... . . . .. . . ..... .. . . . .. .. . . . . . . . . ◦ 2. . . . . . . . . . . . • ... 1. . . .. . • . ..... . . .. . . ... . . . .. . ..... . .. . . ................•............................................................................. ............................................................................................... .. .. . . . . . . 1 3 6 7 What are f (2), f (3), f (6)? Does the function have a limit at x = 2, x = 3, x = 6? Is the function continuous at x = 2, x = 3, x = 6? 28 Speed, rates of change The average rate of change of f (x) over the time interval a ≤ t ≤ b is the difference quotient f (b) − f (a) . b−a The instantaneous rate of change of f (x) at time t is f ( x) − f ( t ) , x→t x−t lim or, equivalently, f (t + h) − f (t) lim , h→0 h 29 Rates of change check your understanding f (b) − f (a) is the slope of the line b−a . 1. The average rate of change through the points and 2. The instantaneous rate of change lim of the f (t + h) − f (t) , is the slope h→0 h line through the point . 30 Rates of change check your understanding 1. The average rate of change f (b) − f (a) is the slope of the line b−a through the points (a, f (a)) and (b, f (b)). f (t + h) − f (t) 2. The instantaneous rate of change lim , is the slope h→0 h of the line through the point . 30-a Rates of change check your understanding f (b) − f (a) 1. The average rate of change is the slope of the line b−a through the points (a, f (a)) and (b, f (b)). f (t + h) − f (t) , is the slope h→0 h of the tangent line through the point (t, f (t)). 2. The instantaneous rate of change lim 30-b Example Let f (t) = t2. The average rate of change from t = 0 to t = 2 is f (2) − f (0) 22 − 02 = = 2. 2−0 2−0 The instantaneous rate of change at t = 0 is f (t + h) − f (t) (t + h)2 − t2 lim = lim h→0 h→0 h h (t2 + 2th + h2) − t2 = lim h→0 h 2th + h2 = lim h→0 h = lim 2t + h = 2t. h→0 31 Study hard and Good luck! 32 ...
View Full Document

This note was uploaded on 03/28/2011 for the course MATH 10A taught by Professor Arnold during the Winter '07 term at UCSD.

Ask a homework question - tutors are online