You may have used shortcuts to see if a number is divisible by another. example "is a number is divisible by 4 if the last two digits form a number divisible by 4" or "a number is divisible by 8 if the last three digits form a number divisible by 8." Where do they come from? Why are some numbers left out? Could you prove these shortcuts?

1. Hallar dos vectores unitarios y ortogonales a 〈?, ?, ?〉 y 〈−?, ?, ?〉.

2. Hallar la ecuación vectorial y ecuaciones simétricas de la línea paralela a la línea que pasa por (−?, ?, ?)y es paralela a la línea ?? = ?? = ? + ?.

how do you know whether you can use a sum or difference identity to determine the exact value of a trigonometric ratio of a given angle?

What strategies do you have for remembering the sum and difference formulas?

A triangle ABC has sides AB=50 and AC=10. D is mid-point of AB and E is mid-point of AC. Angle bisector AG from vertex A meets side BC at G and divides it in 5:1 ratio. So BG=5 times GC. AG cuts ED at F. Find the ratio of the areas of the trapeziods FDBG to FGCE.

1. A shipping company offers various sized shipping boxes to its customers. Some of these boxes are cube-shaped, with equal height, width, and depth. As part of an upcoming sales promotion, the company will offer two cube-shaped boxes for the price of one.

a. Write an expression to represent the total volume of two different sized boxes as a sum of cubes if one of the boxes has sides with a length of 1 foot and the other has sides with a length of x feet.




b. Factor the sum of cubes.




c. Calculate the total volume of the two boxes if x = 3 feet.




2. A toy manufacturer is preparing to manufacture a puzzle cube in two different sizes. One of the puzzle cubes will have a side length of 4 centimeters, and the size of the other is yet to be determined.

a. Write an expression to represent the total volume of two different sized puzzle cubes as a sum of cubes, using m centimeters as the side lengths of the second puzzle.




b. Factor the sum of cubes.




c. Calculate the total volume of the two puzzle cubes if m = 5 centimeters.

3. A pet supply store is planning to offer cube-shaped fish tanks. The store manager decides to order one tank with a 12-inch side length but has not yet decided on the size of the other.

a. Write an expression to represent the total volume of two different sized tanks as a sum of cubes, using 2t centimeters as the length of the sides in the second tank.




b. Factor the sum of cubes.




c. Calculate the total volume of the two fish tanks if t = 7 inches.





4. Think of another real-world situation involving two different sized cube-shaped items.

a. Write a sentence to describe what the cubes represent and then write an expression to represent the total volume of the two items as a sum of cubes, using 2a to represent the side lengths in one cube and 3b to represent the side lengths in the other cube.




b. Factor the sum of cubes.




c. Choose two different values for a and b and use these values to calculate a specific total volume for the two items.

Find the limits, if they exist, or type DNE for any which do not exist.

lim(x,y)→(0,0) (3x^2/(5x^2+4y^2))

1) Along the xx-axis:
2) Along the yy-axis:
3) Along the line y=mxy=mx :
4) The limit is:

Find a vector perpendicular to the line 2y=5-3x in the xy-plane.