Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Corecursion wikipedia , lookup
Knapsack problem wikipedia , lookup
Pattern recognition wikipedia , lookup
Generalized linear model wikipedia , lookup
Inverse problem wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Mathematical optimization wikipedia , lookup
Simplex algorithm wikipedia , lookup
Eigenstate thermalization hypothesis wikipedia , lookup
Continuous-repayment mortgage wikipedia , lookup
MATH 495 Problem Set 1 1. Every point on the parabola y 2 x 1 is equidistant from the y-axis and what other point? 2. If the polynomial px is divided by x 1 the remainder is 1, and if it divided by x 1 the remainder is -1. Find the remainder when px is divided by x 2 1 . 3. If the sinh x 4. x 2 6x 8 If f x x 3 2 x 2 2 x 4 if x 2 is continuous find the value for k. k if x 2 5. What is the domain of the function arcsin ln x ? 6. sinh x e x ex e x ex , cosh x and tanh x find a formula for tanh 1 x . cosh x 2 2 x 2 x 3 if x 1 For what values of m and b will the function f x be differentiable for all values mx b if x 1 of x? 7. What is the slope of the tangent line to the curve xyx y x y 4 at the point 1,1 ? 8. If f x 2 x 1 x 1 what is the value of f ' 0 ? 9. At what value of x in the interval 0, 32 does the function f x 10. How many zeros are at the end of the number 100!? 11. If L is the least common multiple of 1001 and 10101, which of the numbers 6,11,17,22, and 33 is the sum of the digits of L? Can this be determined without actually computing L? 12. Let x1 and x 2 be the two smallest positive integers so that the expression 85x1 12x2 is a multiple of 19. Find x1 x2 . 13. Let G be a group generated by the elements x and y that have the following relations: x 2 y 3 , y 6 1 , and x 1 yx y 1 . Express in simplest form the inverse of the element z x 2 yx 3 y 3 . 14. Let R be a ring; an element x in R is said to be idempotent if x 2 x . How many idempotent elements does the ring ℤ20 contain? 15. 2 2x x sin t t dt attain its maximum value? 1 2 3 If r is the rank and d is the determinant of the matrix 4 5 6 what is r-d? 7 8 9 16. 17. 18. 7 5 For what value of x will the matrix 8 0 20. 0 x 0 1 1 0 not be invertible? 1 1 a Let the vector v = b in ℝ3 and the linear operator 𝑇: ℝ3 → ℝ3 is given by T(u)=vx where denotes c the cross product of the vectors. Then T(u)=Au for a matrix A. Find A. Let y f x be a solution of the equation f 1 ? 19. 6 4 7 0 dy x2 such that y 0 when x 0 . What is the value of 2 dx x 1 A population of bacteria grows at a rate proportional to the number present. After two hours, the population has tripled. After two more hours elapse, the population has increased by a factor of k. What is the value of k? Every curve in a certain family, y f x, C , has the following property: The area of the region in the first quadrant bounded above by the curve from 0,0 to x,0 and bounded below by the x-axis is 13 the area of the rectangle with vertices at 0,0 and x, y . Find f x, C .