Rate of Change

# Rate of Change - -0.1 0.99837 0.01 0.99998-0.01 0.99998...

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2.1 Rate of Change Instantaneous rate of change in the limit of the change in the limit of the average rates of change Which is the slope of the tangents line Ex. With an initial deposit of \$500 balance in a bank account after t years is f(t) = 500(1.05)^t dollars a) Find the average rate of change over [0,1] n f = f(1)- f(0) = 500 (1.05)^1 – 500 (1.05)^0 =25 n t = 1-0=1 n f/n t= 25/1=25 dollar/ year Left hand interval Time interval n t/ n t [0.9,1] 25.552 [0.99, 1] 25.609 [0.999,1] 25.614 [0.9999,1] 25.615 Right hand interval Time interval n t/ n t [1,1.1] 25.677 [1.101] 25.621 b) What is the unit of rate of change of f(t) dollar/ year c) Estimate the instantaneous rate of change at t =1 So the instantaneous rate of change is approximate to 25.615 2.2 limits: A numerical and graphical approach f(x) = sin x / x at x = 0 , f(0)= sin 0 / 0 = 0/0 (undefined) x Sin x/ x x Sin x/ x 1 0.998 -1 0.84147 0.1 0.9998

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Unformatted text preview: -0.1 0.99837 0.01 0.99998-0.01 0.99998 0.001 0.999998-0.001 0.99999833 Definition of limit: Assume f(x) is defined for all x near c. but not necessarily at c. its e(t) we say the limit of f(x) as x approach c is equal to L. If f(x)-4 becomes arbitrarily small when x is any number sufficiently close to c (not equal) We write: Lim f(x) =L We say f(x) converges to L as x ->c Ex. Lim (x-:>2) 2x+3 = 7 Absolute value of f(x) – L = absolute value of 2x+3 -7 = absolute value of 2x-4 = 2 times the absolute value of x-2 Since absolute value of f(x) – L is a multiple of the absolute value of x-2, Absolute value of f(x) – L become arbitrary small as x is close to 2. Theorem a) lim x->2 k=k b) lim x->c x= c ex. Lim x->0 e ^(-1/x^2) = 0 x F(x) + 1 e ^-1 +0.1 e ^-100 +0.01 Example 1 (limit equals function value) F(x)= x Lim (x->4) x = 2...
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Rate of Change - -0.1 0.99837 0.01 0.99998-0.01 0.99998...

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