Math Solutions using Sigma Notation, Integration, and Graphical Representation

1. Express the sum 22 + 33 + 44 + 55 + ··· + 2121 using the sigma notation.

 

2. Develop and evaluate

3.You must use the properties of the sigma notation.

 

4. Evaluate (k + 3). You must use the properties of the sigma notation.

 

5.Knowing that. Evaluate .

6. For which integer and positive value of not do we have that k = 105?

 

7.Julie estimated that she would spend an average of $ 2000 per year on the purchase of broccoli over the next 15 years. She wants to determine the amountM which she would have in 15 years if, at the end of each year, she deposited this sum in a bank account bearing an annual interest rate of 3%. The mathematical formula to calculate this value is or R represents the annual deposit, i represents the annual interest rate and not the number of deposits made.

• Check that in general, R(1 + i)k?1 = R not ( ) (1 + i)not - 1 i

• (3 points) Calculate the amount M that Julie will have in her account after 15 years.

3.  We construct a regular partition of the interval [-5, -1] defining not subintervals: - 5 = x0 < x1 < x2 < x3 < ··· < xk?1 < xk < ··· < xn?1 < xnot = -1.

(a) (3 points) Represent the surface graphically S delimited by the equation curve y = x2 over the interval [-5, -1].

(b) What is the larger of x?

• What is the expression of xk in this score?

• We can show that the sum Snot areas of not rectangles which are erected on the subintervals determined by the partition and which circumscribe the surface S is given by n??1( ) 100 not 160k not2 64k2 not3 Snot = - + . k= 0 Use the properties of the sigma notation to show that 80 (n - 1) not 32 (n - 1) (2n - 1) 3not2 Snot = 100 - + .

• From the answer obtained in (d), evaluate the exact area of the surface S using a limit process.

• Check that 1/3x^3 is a primitive of x2.

• Use the fundamental theorem of integral calculus to confirm the result obtained by (e).

4. Either f(x) and g(x) functions that can be integrated on R and such as

(a) (4 points) What is 3f(x) dx ?

(b) (4 points) What is

(c) (2 points) What is (f(x) + 3g(x)) dx ?  Check that ln (3x2) is a primitive of 2/x

6. Evaluate the definite or indefinite integral.