Calculus I Homework: The Derivative as a Function Page 1
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1 Calculus I Homework: Te Derivative as a Function Page 1 Example (2.9.16) Make a careful sketc of te grap of f(x) = sin x and below it sketc te grap of f (x). Try to guess te formula of f (x) from its grap. 1) Tis is neat! I ve sketced te sine function, y = f(x) = sin x. Te key idea we use over and over again ere is tat te value of te derivative at x = a is equal to te slope of te tangent line to te curve at x = a. At te points were te sine function is ±1 it as a orizontal tangent. Tis means tat te derivative f (x) must be zero at tose points. So f (π/2) = f (3π/2) = 0. Te sine function as a minimum slope at x = π, and if you look closely (or create a new grap by zooming in) you can guess tat te value of tat minimum slope is 1. Tis means tat f (π) = 1. Te sine function as a maximum slope at x = 0 or 2π, and if you look closely (or create a new grap by zooming in) you can guess tat te value of tat minimum slope is +1. Tis means tat f (0) = f (2π) = 1. For 0 < x < π/2, te sine function is increasing. Tis means its derivative will be positive in tis region. For π/2 < x < 3π/2, te sine function is decreasing. Tis means its derivative will be negative in tis region. For 3π/2 < x < 2π, te sine function is increasing. Tis means its derivative will be positive in tis region. And tat says in words every tougt and relation tat we used to construct te sketc of te derivative of te sine function, wic is included on te next page. Te derivative sure looks like te cosine function!
2 Calculus I Homework: Te Derivative as a Function Page 2 Example (2.9.18) Make a careful sketc of te grap of f(x) = ln x and below it sketc te grap of f (x). Try to guess te formula of f (x) from its grap. 2) Again, te key idea we use ere is tat te value of te derivative at x = a is equal to te slope of te tangent line top te curve at x = a. First, let s sketc te logaritmic function y = f(x) = ln x.
3 Calculus I Homework: Te Derivative as a Function Page 3 Tis one isn t as nice as te sine function we just finised, since tere are fewer points to elp guide us. We could zoom in on a bunc of points, work out te slope (since te grap will appear as a straigt line wen we zoom in) and ten plot tose values as points on te derivative curve. Joining te points would give us a grap of te derivative f (x). Instead, let s do wat te problem suggests, and look at te grap in broader scope. First, te logaritmic function is always increasing, so te derivative function will always be positive. Second, as te logaritmic function approaces zero from te rigt, it grows infinitely large negative. Tis means our derivative function sould blow up at x = 0, and since it as to be positive, it must approac positive infinity. So we know tat lim x 0 + f (x) =. Te last ting we can say from looking at te grap of te logaritmic function is tat te slope of tangent to te curve is decreasing. Tis means f (x) is a decreasing function of x. I ve included all tese details in te sketc below. It seems to me tat we may ave f (x) = 1 x.
4 Calculus I Homework: Te Derivative as a Function Page 4 Example (2.9.47) Prove eac of te following: a) Te derivative of an even function is an odd function. b) Te derivative of an odd function is an even function. a) Here, we need to work from te definition of derivative. f f(x + ) f(x) (x) = lim. If te function f(x) is even, ten f( x) = f(x). To sow tat te derivative is going to be odd, we need to sow f ( x) = f (x). f f( x + ) f( x) ( x) = lim f(x ) f(x) = lim since f is even
5 Calculus I Homework: Te Derivative as a Function Page 5 Now we can t jump to our conclusion too quickly! Tis doesn t quite look like te definition of derivative we know and love. We really want f(x + ) in tere. So let s make a substitution =, and see wat appens. As 0, so will 0. f ( x) = lim = lim = f (x) So we ave proven tat if f is even, ten f is odd. We will see anoter way of proving tis statement once we ave learned te cain rule for derivatives. b) Here, we need to work from te definition of derivative. f f(x + ) f(x) (x) = lim. If te function f(x) is odd, ten f( x) = f(x). To sow tat te derivative is going to be even, we need to sow f ( x) = f (x). f ( x) = lim 0 f( x + ) f( x) = lim 0 f(x ) + f(x) = lim 0 f(x ) f(x) since f is odd since f is odd Just like above, we need to make a substitution before we are done. Let s make te substitution =, and see wat appens. As 0, so will 0. f ( x) = lim = + lim = +f (x) So we ave proven tat if f is odd, ten f is even.
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