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MATH 117 (2)

Math 117 Fall 2012 Course Notes

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University of Waterloo
MATH 117
Eddie Dupont

MATH 117 - Calculus 1 for Engineers Kevin Carruthers Fall 2012 Functions A function is a rule which assigns a single output to a variable given at least one input variable. In this class we deal only with a single input x and write y = f(x) x is the indepenant variable y is the dependant variable A domain is the set of allowable values for the independant variable (x) A range is the set of possible values for the dependant variable (y) Function Composition If y = f(x) and x = g(t) then we can view y as a function of t, y = f(g(t)), which we sometimes write as y = f g(t) 3 For f(x) = 1 sinx;g(x) = x , f g(x) = f(x ) = 1 sinx3 3 g f(x) = g(1 sinx) = (1 sinx) Domain of Composite Functions For f g(x), R = fy j y = f g(x); x Dgg or R = fy j y = f(x); x R g g 1 Inverse Functions We say g is the inverse of f if g(f(x)) = x for all x in the domain of f. If g(f(x)) = x, then Rf D g Typically, the inverse of f(x) is written as 1 g = f (x) Note that this does not mean the reciprocal: 1 1 f(x) = (f(x)) Finding Inverses In simple cases, we just solve y = f(x) for x. Example: if f(x) = x2 , nd the inverse x 1 y = 2 2y = x 1 2y + 1 = x f 1(y) = 2y + 1 Does f 1(x) always exist? No, f(x) only has an inverse if it is one-to-one, ie. if its graph passes the horizontal line test. Mathematically, if f(x0) = f(x1); 0 = x1 We can restrict the domain so a non-invertible function has an inverse: p f(x) = x on [0;1) has inverse f 1(x) = x 2 1 p f(x) = x on (1;0] has inverse f (x) = x 2 Even and Odd Periodic Functions A function is even if f(x) = f(x), this means its graph is symmetric about the y-axis. A function is odd if f(x) = f(x), this means its graph is symmetric about the origin (ie. re ected in the x- and y-axes). Most functions are neither even nor odd, but we can transform any function into the sum of even and odd parts. f(x) f(x) f(x) = f(x) + 2 2 f(x) f(x) f(x) f(x) = + + 2 2 2 2 = f(x) + f(x)+ f(x) f(x) 2 2 = g(x) + h(x) where g(x) is an even function and h(x) is an odd function. This idea is used to form two useful functions. x e + ex e ex e = 2 + 2 = coshx + sinhx Usefulness of Symmetry To sketch f(x) = x jxj, we note that f(x) is even, thus we can consider only x > 0 and use symmetry to determine x < 0 Periodicity A function f(t) is periodic with period P if f(t + nP) = f(t); n Z Any function which exists only over a nite interval can be extended as an even or odd periodic function. Absolute Values ( jxj = x if x > 0 x if x < 0 3 p 2 p 2 p 2 Alternatively, we can use jx . Note: x 6= x. Note 2: ( x) = x Geometrically, jxj is the distance between x and 0 on the number line. Similarly, jx aj is the distance between x and a on the number line. Heavyside Function ( 0 if t < 0 H(t) = (1 if t 0 H(t a) = 0 if t < a 1 if t a Multiplying a function by H(t a) is like a switch which "turns the function on" at t = a. We can combine multiple Heavyside functions to "activate" certain functions in a specied region. Example: f(t) = t + (2 t)H(t + 2) + (t + 1)H(t) + (2 t t )H(t 1) 8 >t if t (1;1) <2 if t [2;0) f(t) = 2 >t + 1 if t [0;1) :3 t if t [1;1) How do we do this in reverse? ie: take a piecewise function and denite it in terms of H(t a)? 1. Use the fact that (for a < b) 8 > 0 if t < a < H(t a) H(t b) =1 if a t < b : 0 if t b makes an on/o switch. So does ( 1 if t < a 1 H(t a) =0 if t a 2. When a new function is turned on, subtract the previous one. 4
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