Download Study Guide for Test #3 on 11/26

Mathematics 2122-002
Calculus for Life Sciences II
Fall 2013
Study Guide for Test #3
Instructor: Dr. Alexandra Shlapentokh
(1) Let f (x, y, z) = x2 y−3z+5. Compute f (3, 2, 2). Solution: f (3, 2, 2) = 9·2−3·2+5 =
17.
p
(2) Find the domain of f (x, y) = x2 − yx. Solution: {(x, y) ∈ R2 : x2 − yx ≥ 0}.
x2
(3) Find the domain of f (x, y) = yx−1
. {(x, y) ∈ R2 : yx 6= 1}
2
x
(4) Find the domain of f (x, y) = √yx−1
.Solution: {(x, y) ∈ R2 : yx − 1 > 0}
(5) Compute ∂f /∂x, ∂f /∂y, ∂ 2 f /∂x2 , ∂ 2 f /∂y 2 , ∂ 2 f /∂x∂y for the following functions:
f (x, y) = x ln y + x sin y − exy Solution:
∂f /∂x = ln y + sin y − yexy ,
∂f /∂y =
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
x
+ x cos y − xexy ,
y
∂ 2 f /∂x2 = −y 2 exy ,
x
∂ 2 f /∂y 2 = − 2 − x sin y − x2 exy
y
1
∂ 2 f /∂x∂y = ∂ 2 f /∂y∂x = + cos y − yxexy − exy ,
y
What is a differential equation? (See Definition, page 546)
How can one determine the order of a differential equation? (See Definition, page
546)
What is a general solution to a differential equation? (See page 547)
What is a particular solution to a differential equation? (See page 547)
What is an initial condition? (See page 547)
What is an initial value problem? (See page 547)
Is f (x) = sin x + ex a solution to the differential equation y 00 + y = ex ? (No)
Is f (x) = sin x − ex a solution to the differential equation y 00 + y = ex ? (No)
Is f (x) = ex a solution to the differential equation y 00 + y = 2ex ? (Yes)
What is a general solution to the equation y 0 = f (x), where f (x) is a continuous
function?
R
y = f (x)dx
Find a particular solution to the equation y 00 = x3 , if y 0 (1) = 0 and y(2) = 0.
4
We integrate y 00 to conclude that y 0 = x4 + C. Substituting 1 for x and 0 for y we
4
obtain 0 = 14 + C and conclude C = − 14 . Thus y 0 = x4 − 14 . Integrating y 0 we
5
32
obtain y = x20 − 14 x + C. Substituting 2 for x and 0 for y we obtain 0 = 20
− 21 + C.
5
11
11
Therefore, C = − 10
and y = x20 − 14 x − 10
.
What is a linear first-order differential equation? (Definition, p. 552)
1
(18) When is a linear first-order differential equation homogeneous? (Definition, p.
552)
(19) Consider a differential equation of the form y 0 + p(x)y = q(x), y(x0 ) = y0 , where
p(x) and q(x) are continuous in some interval I containing x0 . What conclusion
can one draw about a solution to this equation? (See Theorem 2, p. 552)
(20) Describe a general solution to a differential equation y 0 + p(x)y = q(x). (See
Theorem 3, page 554.)
(21) Examples 3, 4, page 554-555
(22) Solve xy 0 − 4y = x, y(2) = 3
R
Rewrite the equation y 0 − x4 y = 1. Note that p(x) = − x4 , F (x) = − x4 dx = ln x−4 +C,
−4
G(x) = eln x = x−4 . Next we have (yx−4 )0 = x−4 , yx−4 = − 31 x−3 + C, y =
− 13 x + Cx4 . Substitute y = 3, x = 2 to get 3 = − 23 + 16C, C = 11
. Final answer is
48
11 4
1
y = − 3 x + 48 x .
(23) Examples 5 – 7, pages 557-560
(24) Suppose a tank contains 200 gal of water and 7 pounds of a certain chemical. A
solution of the same chemical containing 3 pounds per gallon is being poured into
the tank at the rate of 5 gallons an hour while the tank is being drained at the same
rate. Let A(t) be the amount of the chemical in the tank at time t. Determine the
formula for A(t) and limt→∞ A(t). (Assume the distribution of the chemical in the
tank is uniform at any moment of time.)
(25) Separable differential equaltions: Example 1, 2, 3, 5 pages 575 – 579
(26) What is the general solution to the equation y 0 = ky?
(27) Solve y 0 = 0.1y, y(1) = 2.
(28) Solve y 0 = 3xy, y(0) = 5.
3
(29) Solve y 0 = 2et t2 y.
et
(30) Solve y 0 = 2
y +y
(31) Higher Order Homogeneous Differential Equations: Examples 1, Example 5, Example 8, pages 598–602
(32) Solve y 00 − 5y 0 − 6y = 0, y 0 (0) = 1, y(0) = 1
(33) Solve y” − 2y 0 + 1 = 0, y 0 (0) = 1, y(0) = 1
(34) Solve y 00 + 16y = 0.
2