WS21 110420 - Worksheet PDF

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Math 221 Worksheet 21 Wednesday November 4th 2020

Definite Integrals and the Fundamental Theorem of Calculus (1) Assume that f is a continuous function. The definition of the definite integral Z b n X f (x) dx = lim f (xi ∗ )∆x. a

Rb f (x) dx is a

n→∞

i=1

Answer the following questions about this definition. You will probably need to use both words and mathematical symbols in your answers. (a) What is meant by n? (b) What is meant by ∆x? (c) What is meant by xi∗ ? n n X X i2 n3 n2 n 2 using the fact that + . + i = n→∞ 6 n3 2 3 i=1 i=1 What definite integral might this represent?

(2) Compute lim

n X n2 n i i= ) using the fact that + . n→∞ n 2 n 2 i=1 i=1 What definite integral might this represent?

(3) Compute lim

n X 1

(1 +

(4) Suppose that f is an odd Z 3function and g is an Z 3even function which are each integrable on the interg(x) dx = 5, evaluate the following definite integral. val [−3, 3]. Given that f (x) dx = 4 and 0 0 Z 3 (6f (x) + 8g(x)) dx −3

1

(5) Let f be the function whose graph is shown below and let g(x) =

Rx 0

f (t)dt.

(a) Find g(0), g (1), g (2), g(3), g (4), g (5), g(6). (b) Estimate g(7). (c) Where does g have an absolute mininum and absolute maximum on [0,7]? (d) Where does g have local extrema, if any? (e) Where is g concave up and where is g concave down?

(6) Let F (x) =

Z

x

sin(t2 + 5) dt. Find F ′ (x).

0

(7) Let G(x) =

Z

x2

(t − 169)92 (5t − 245)37 dt. Determine the x-value for each critical point of the

−19

graph of G(x).

2...