X'0, X1,* , Xn E (a, b) are arbitrary points such that x; # x; for i # j. Fur- thermore, let P be a polynomial of degree n such that P(x;) = f(x;) for all i = 0, 1, · .. ,n. Prove that for every x E (a, b) there exists ( such that min{x, xo, x1, .. , Xn} < $ < max{x, xo, x1, , xn}, and %3D f(n+1)(C) f(x) – P(x) = (x – xo)(x – x1) ·.· (x – xn). (n + 1)! | | Hint: Fix x, and denote by M the number that satisfies the equation M ;(x – xo)(x – x1) · · (x – xn). (n + 1)! f(x) – P(x) : %3D ...

Matching functions and derivatives Match the functions a-d with the derivatives A-D. yA Functions a-d yA 4 3+ 3 2+ 1+ 1+ 1 3 4 1 3 4 (a) (b) YA YA 4+ 3+ 3+ 2+ 2+ 1+ 1+ + 3 4 x + + 3 2 4 (c) (d) 2. 1 2. 2. 2. 1,

Calculus I May I please have an explanation and concept for the criteria given?   Thank you so much

6. AVERAGE RATE OF CH ANGE The average rate of change of the function y = f(x) between x = a and x = change in y f(b) – f(a) b – a average rate of change change in x The average rate of change is the slope of the secant line between x = a and x = 1 on the graph of f, that is, the line that passes through (a, f(a)) and (b, f(b)). f(b) – f(a) b- a yA average rate of change f(b)- y = f(x) f(b) - f(a) fla)t 0. a Using the information above, find the average rate of change of the following functions in the given interval: a) **Let f(x)= 3x – 5. Find the average rate of change of f(x) on the interval [0, 1]. Then find it on the interval [3, 7]. Please explain your answer. You can use the graph for the explanation. b) Let f(x)=(x – 3)². Find the average rate of change of f(x) on the interval [1, 3]. Then find it on the interval [4, 7]. Please explain your answer. You can use the graph for the explanation.

Two part question