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Please pick between one of the 2 screenshots.CHAPTER 2 MAKING CONNECTIONS
1. Suppose that is a function with the properties (1) f is differentiable everywhere, (ii) f(x+y)=f(x)f(y) for all values of y and y, (iii) f (0) #0, and (iv) f'(0)=1.
(a) Show that f(0) = 1.
[Hint: Consider f (0+0). ]
6) Show that f (x) > 0 for all values of x-
[Hint: First show that f(x) + for any x by considering f(x – x).]
© Use the definition of derivative (Definition 2.2.1) to show that f'(x) = f (x) for all values of x-
2. Suppose that f and g are functions each of which has the properties (1)-(iv) in Exercise 1).
(a) Show that y = f (2x) satisfies the equation y’ = 2y in two ways: using property (11), and by directly applying the chain rule (Theorem 2.6.1).
6) If he is any constant, show that y = f (lox) satisfies the equation y’ = ky
© Find a value of such that y = f(x)g(x) satisfies the equation y’ = ky-
(d) Ifh= fg, find h'(x). Make a conjecture about the relationship between f and g.
3. (a) Apply the product rule (Theorem 2.4.1) twice to show that if f.g, and h are differentiable functions, then f g h is differentiable and
(f.g. h)’ = f’.g.h+f.8.h+f.g.h’
6) Suppose that f:g, h, and fe are differentiable functions. Derive a formula for (f.g.h-k)’
©) Based on the result in part (a), make a conjecture about a formula differentiating a product of n functions. Prove your formula using induction.
4. (a) Apply the quotient rule (Theorem 2.4.2) twice to show that if f, g, and h are differentiable functions, then (f/g)/h is differentiable where it is defined and
[(f/g) /h’ =
figh-figh-figh!
Ph2
f’isch-
6) Derive the derivative formula of part (a) by first simplifying (f/g)/h and then applying the quotient and product rules.
©) Apply the quotient rule (Theorem 2.4.2) twice to derive a formula for f (g/h)]’
(d) Derive the derivative formula of part (c) by first simplifying f / (g/h) and then applying the quotient and product rules.
5. Assume that h(x) = f(x) g(x) is differentiable. Derive the quotient rule formula for h'(x) (Theorem 2.4.2) in two ways:
(a) Write h(x) = f (x) – [g(x)] – and use the product and chain rules (Theorems 2.4.1 and 2.6.1) to differentiate h-
6) Write f (x)=h(x)-g(x) and use the product rule to derive a formula for n'(x)-
CHAPTER 3 MAKING CONNECTIONS
In these exercises we explore an application of exponential functions to radioactive decay, and we consider another approach to computing the derivative of the natural exponential function.
1. Consider a simple model of radioactive decay. We assume that given any quantity of a radioactive element, the fraction of the quantity that decays over a period of time will be a constant that depends on only the particular element and the length of the time period. We choose a time parameter – 0, you can interpret A(t) as the fraction of any given amount that remains after a time period of length 1.]
(6) As+t) = 4(5) At)
[Hint: First consider positive s and 1. For the other cases use the property in part (a).]
© If n is any nonzero integer, then
4() = (A(1))}’n = öl/
(d) If m and n are integers with n= 0, then
4(17) = (4(1))m’n = 5min
e) Assuming that Act) is a continuous function of , then A(t)=
[Hint: Prove that iftwo continuous functions agree on the set of rational numbers, then they are equal.]
(f) If we replace the assumption that 4(0) = 1 by the condition 4(0) = 4o, prove that A=4054
2. Refer to Figure 1.3.4.
(a) Make the substitution h=1/x and conclude that
(1+h)1/h
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