For Exercises 1-2, for parts (a) and (b), graph the equation. For part (c), solve the system of equations. For parts (d) and (e) graph the solution set to the system of inequalities. If there is no solution, indicate that the solution set is the empty set. a. y = − 3 x + 5 b. − 2 x + y = 0 c. y = − 3 x + 5 − 2 x + y = 0 d. y > − 3 x + 5 − 2 x + y < 0 e. y < − 3 x + 5 − 2 x + y > 0
For Exercises 1-2, for parts (a) and (b), graph the equation. For part (c), solve the system of equations. For parts (d) and (e) graph the solution set to the system of inequalities. If there is no solution, indicate that the solution set is the empty set. a. y = − 3 x + 5 b. − 2 x + y = 0 c. y = − 3 x + 5 − 2 x + y = 0 d. y > − 3 x + 5 − 2 x + y < 0 e. y < − 3 x + 5 − 2 x + y > 0
Solution Summary: The graph for the line y=-3x+5 is: ly = 3(0)+0 0+y
For Exercises 1-2, for parts (a) and (b), graph the equation. For part (c), solve the system of equations. For parts (d) and (e) graph the solution set to the system of inequalities. If there is no solution, indicate that the solution set is the empty set.
1. Show that the vector field
F(x, y, z)
=
(2x sin ye³)ix² cos yj + (3xe³ +5)k
satisfies the necessary conditions for a conservative vector field, and find a potential function for
F.
1. Newton's Law of Gravitation (an example of an inverse square law) states that the magnitude
of the gravitational force between two objects with masses m and M is
|F|
mMG
|r|2
where r is the distance between the objects, and G is the gravitational constant. Assume that the
object with mass M is located at the origin in R³. Then, the gravitational force field acting on
the object at the point r = (x, y, z) is given by
F(x, y, z) =
mMG
r3
r.
mMG
mMG
Show that the scalar vector field f(x, y, z) =
=
is a potential function for
r
√√x² + y² .
Fi.e. show that F = Vf.
Remark: f is the negative of the physical potential energy, because F = -V(-ƒ).
2. Suppose f(x) = 3x² - 5x. Show all your work for the problems below.
Elementary Statistics: Picturing the World (7th Edition)
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