13 Tutorial Introduction to Calculus and Linear Algebra
Tutorial 13 Introduction to Calculus and Linear Algebra
In Exercises 1 to 8, find the indicated integral. 1.
3.
5. 7.
(x
0
1
4
+ 3 x 3 + 1)dx
2.
4. 6. 8.
1 + x + x
1
3
1
1 dx 2
-1
-3 1
t +1 dt t3
3
2
1
(2 x - 4) 4 dx
(x
0
+ x) x 4 + 2 x 2 + 1dx
e2
1
(ln x) 2 dx x
x dx x -1 1/ x 1/ 2 e 1/ 3 x 2 dx
e +1 2
In Exercises 9 to 11, find the area of the region R that lies under the given curve y = f(x) over the indicated interval a < x < b. Under y = x 4 , over 1 < x < 2 11. Under y = (3 x + 4)1/ 2 , over 0 < x < 4 9. 10. Under y = e 2 x , over 0 < x < ln 3
In Exercises 12 to 15, find the area of the region R. 12. 13. 14. 15.
R is the region bounded by the lines y = x, y = x, and x = 1. R is the region bounded by the x axis and the curve y = - x + 2 4 x - 3 . R is the region bounded by the curve y = x 2 - 2 x and the x axis. R is the region bounded by the curves y = x 2 - 2 x and y = - x 2 + 4 .
In Exercise 16 to 21, solve the first order ordinary differential equations. dy 17. y ' = xy 2 ; y(1) = 5 16. = 2 x(1 + y ) dx ex ; y(0) = 1 19. x 2 y '+3 xy = 4 x 2 + 2 18. y ' = y 20. y ' =
xe x ; y(0) = 0 1 - y5
2
21. Let A(t) denote the amount of a radioactive substance that is present in a sample at time t. d The usual model for radioactive decay is that A(t ) = kA(t ) . The rate of decay for dt radioactive isotopes is usually specified in terms of the half-life. The half-life of an isotope is the time required for one-half of the initial amount to decay. If the half life of an isotope is 12 years, what is k? What is the general relationship between the half-life of the isotope and k?
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