Question 3. Evaluate (-t, t, -2t)|| dt.

6. (2x +3y = 0 /2x x+2y =-1 9. (1 1 -X+-y 5 1, -x+y%3D10 4

3) Personal taxes Suppose that the percent of total personal income that is used to pay personal taxes is given by y = 0.034x² – 0.044x + 12.642 where x is the number of years past 1990 (Source: Bureau of Economic Analysis, U.S. Department of Commerce). (a) Verify that x = 7 is a solution of 14 = 0.034x² – 0.044x + 12.642. (b) Find the year or years when the percent of total per- sonal income used to pay personal taxes is 14.

solve the solution ;

1. Verify that the formulas for Bo and B1 for linear equation ŷ = Bo + B1x from A = (X" X)-1 X" Y are the same as those formulas that we mentioned before. That is ( is short for Σ) nEriy; – (Ex;)(E4%) B1 Ex? – nữ? n Bo = – B1ữ XiYi (E4i)(E) – (Dr;)ET:y; nE – (E#)²

Hermann Ebbinghaus (1850–1909) pioneered the study of memory. A 2011 article in the Journal of Mathematical Psychology presents the mathematical model R(t) = a + b(1 + ct)−? for the Ebbinghaus forgetting curve, where R(t) is the fraction of memory retained t days after learning a task; a, b, and c are experimentally determined constants between 0 and 1; ? is a positive constant; and R(0) = 1. The constants depend on the type of task being learned. (a) What is the rate of change of retention t days after a task is learned?

Answer No. 1 and 2

In Problems 1 through 6, show directly that the given functions are linearly dependent on the real line. That is, find a non- trivial linear combination of the given functions that vanishes identically. 1. f(x) = 2x, g(x) = 3x², h(x) = 5x – 8x² 2. f(x) = 5, g(x) = 2 – 3x², h(x) = 10 + 15x2 3. f(x) = 0, g(x) = sin x, h(x) = e* 4. f(x) = 17, g(x) = 2 sin² x, h(x) = 3 cos? x 5. f(x) = 17, g(x) = cos² x, h(x) = cos 2x 6. f(x) = e*, g(x) = cosh x, h(x) = sinh x %3D %3D

12. Is the data x representative of a linear, quadratic, or exponential model? x -3 -2 -1 0 -18 -6 -2 0 y O 1 2 2 3 6 18 linear quadratic exponential -3 -2 -2 -1 -1 0 1 y 18 -6 -2 0 2 * 1 2 6 × 18