Find the geneml solution of the given differential equation.
+ 1 Eli + + 2 - 2x 1
(X ) dx cfw_X )y e
x2 c
r- +
(x+1)9x (x+1)ex
V
Give the largest interval 1 over which the geneml solution is dened. (Think about the implications of any singular poi
Practice Test 1 (MATH 165 Butler)
Student name:
This test is closed book and closed notes. No sophisticated calculator is allowed for this test. For full
credit show all of your work (legibly!). If you know about LHospitals method, then do not use it to
a
Practice Test 1 (MATH 165 Butler) Strident name:
This test is closed book and closed notes. No sophisticated calculator is allowed for this test. For full
credit show all of your work (legiblyl). If you know about LHospitals method, then do not use it t
Find an equation fertne |ine tangentte the curve at the point dened bytne given value eft. Nee, ndtne
:13?
value of at this point.
m2
1
x=3t,y=1,t=E Assuming thattne equatien denes it andy implicitly as differentiable functions it =f|[t:I, y: git, ndtne
s
Find the Cartesian coordinates of the following points [given in polar coordinates.
21 i":
a. cfw_2,3 b. cfw_11 c. [5,?] d. [4,?] Find the pular vac-ordinates, DEE! < 2: and r2 i], of the following paints given in Carteaian cunrdinatea.
cfw_a (5&5) cfw_hi
Find the Taylor polynomial of order 3 based at a for the function.
32"; a=T
Find the Taylor polynomials of orders [1, l, 2, and 3 generated by f at x =a.
-5 -2
x_xr E. Find the Taylor polynomial of orders [1, l, 2, and 3 generated by f at a.
x=sinx, a=1t.
The equation below gives parametric equations and parameter intervals for the motion of a particle in
the xy-plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian
equation. Indicate the portion of the graph traced
ISU Academic Success Center
Calculus II
Areas and Lengths in Polar Coordinates (11.5)
Problem 1: Find the area inside one leaf of the three-leaved rose r=cos 3
2
Problem 2: Find the area of the region inside the lemniscate r =6 cos 2
the circle r= 3
and o
ISU Academic Success Center
Calculus II
Binomial Series & Parametrizations of Plane Curves
Problem 1: Find the first four terms of the binomial series for the given functions:
a.
(1+ x)
1
3
3
b. (17 x )
Problem 2: Use series to evaluate the limits below:
ISU Academic Success Center
Calculus II
Calculus with Parametric Curves
Problem 1: Find an equation for the line tangent to the curve at the point defined by the
2
given value of t . Also, find the value of
a.
x=t ; y =t ; t=
1
4
d y
dx 2
b.
at this point
ISU Academic Success Center
Calculus II
Polar Coordinates and Graphing in Polar Coordinates (11.3 & 11.4)
Problem 1: Replace the polar equations below with equivalent Cartesian equations.
Then describe the graph.
a.
r=cot csc
b. r=2 cos + 2sin
c. r sin +
Objective: To Solve Optimization Problems
Which is greatest? Which is least? Minimize, Maximize.
Used in Business for Cost, Production, Volume, Area
Environment for Volume, Area, Cost of Pollution
Energy for Fuel Consumption, gas mileage
Traffic Flow, Med
Oblique Asymptotes (€26 maimed)
Let f(x) be a rational function, i.e.,
f<xr= :83)
If deg P(x) = deg S(x) +1 , then the graph of f(x) has an oblique
or slant line asymptote.
This means that, for large magnitude of x, the behavior of f(x)
is similar to
3‘3 Differentiation Ruies
Derivative of a Constant Function f(x)=o
'( =t/i ’VXH‘B'RX): Linn C " C- :hwi O = O
4‘ X) [rm E k-VO k k—vd
”if m: (at; O ‘1
RQWGUUIG for positive integers: Let n be a positive integer,
we’llﬂvfind the derivative of f(x)= xﬂ
Name:
MATH 165 Section_ Exam 1 2/5/2016
SHOW ALL YOUR WORK to avoid loss of points.
tan(5;‘c)
1. Redeﬁne the function f(x) 2 ‘ so that it is continuous at x : 0., and hence for all real cc.
x
[Hint]: For the required lirnit7 express f (:23) in terms
3.2 The Derivative as a Function
Last time we defined the derivative of y=f(x) at the point x: x0
¥\(2¢D= um $(x.+k)~ gm)
la
M-NJ
Definiﬁgn The derivative of the function f(x) with respect
to the variable x is the function f’ whose value at x is:
\ _ 1m 1
2‘6 Limits lnvolvin
lottery
We are interested in the behavior of a function when the
magnitude of the independent variable x becomes
increasingly large, we write: X —+ 00 or X a _ co
plxl loge X <0
* The symbol 00 W represent a number, we use it to
desc
Continuity Test
A function f(x) is continuous at an interior
point x=c of its domain if and only if:
1. f(x) is
2.
3.
We have three kinds of discontinuities:
Discontiniuty
Discontiniuty
Discontiniuty
A function is continuous if it is continuous at every
p
2.4 One-Sided Limits
Definitions: (Side limits) Left-hand side limit; we write
This is the value that f(x) approaches, if it exists, when x approaches c
from the left, that is we consider values
Similarly, right- hand side limit; we write
This is the valu