18.11. If f(z) is analytic and |f(z)| ≤1/(1-2) in || < 1, show that |f'(0)| ≤ 4.
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- What is the purpose of the Intermediate Value Theorem?Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.What does it mean to say that f(4) is a local maximum valueof f ?
- 5. Let f be defined for all real numbers and suppose if(x)-f(y)| ≤ (x - y)² for all real numbers, x and y. Prove f'(y) = 0, vy E R.6. Use the Mean Value Theorem to find all values of c in (6, 14) such that ƒ'(c) = f(b)-f(a) b-a for f(x)=√√√2x - 3 on the interval [6, 14]3 2) Let f(x)= x³ - 2x+. Calculate by definition ƒ'(1). X
- 4. There is a function f that's continuous at 0 and /(x) = {(2x) for all x's please show that f must be a constant function. (1A.) Use induction to show that f(x) =f(2"x) for each integer n>0. (1B.) For a fixed x, how does 2"x behave for large integers n? (1C.) Using A. and B and the assumption about continuity, Explain why f(x)= f(0) for each x.1) In this problem, let x = 2, the value Ax = dx, dy = f'(x)dx, and Ay = f(x + Ax)-f(x) to complete the table f(x)=√x 1.0000 0.5000 0.1000 0.0100 0.0010 2) dx= Ax dy Ay Ay - dy dy Ay 3) 4) f(x) = 5x √2x+3 Find the first derivative given that dy Find X by implicit differentiation. √√x+y+√√x = 4 dx -лr² h A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at the rate of 10 cubic feet per minute, find the rate of change of the depth of the water the instant it is 8 feet deep? 1 V10 5. Consider a function f (x) that satisfies all the conditions listed. i. f is defined and continuous on the interval [-4,4]. f' is defined on the intervals (-4,0) and (0,4). f' is positive on (-1,0) and (2,4); f' is negative on (-4, – 1) and (0,2). f" is positive on (-2,0) and (0,3); f" is negative on (-4, –2) and (3,4). ii. ii. iv. a. Fill out the following table describing the features of f using the information about the signs of f' and f" given above. Behavior of f For x in the interval: (-4,-3) (-3,-2) (-2,-1) (-1,0) (0,1) (1.2) (2,3) (3,4) Increasing/ decreasing Concave up/ concave down