Download review for final - Seattle Central College

Math 152 Review for Final
You may have one page of notes (both sides), and a graphing calculator. Here are the topics on the test
Topic
Sub-topic
Σ (summation)
Computation
Using limit as n approaches infinity to compute an integral (Reimann
sum)
Numeric Integration
Right endpoint, Left endpoint, Midpoint
Trapezoid rule
Simpson’s rule
Integration methods
Basic anti-derivatives (trig, exponential, etc)
u – substitution
integration by parts
partial fractions
trig substitution
using trig identities
table of integrals (the formula will be provided)
Improper integrals
Type 1 and Type 2
Applications of integration
Area
Area between two curves
Length of a curve
Solids of revolution (x-axis, y-axis, washer method, cylindrical
method)
Work (for instance: compressing a spring, pulling something with a
rope, or pumping water)
Area functions and
Fundamental Theorem of
Calculus
Differential equations
Make a graph of an area function
Find derivative on an area function
Be able to justify steps in the proof
Verify a solution to a differential equation (example: show that
y = e-x/2 is a solution to 2y´ + y = 0)
Find the value of a constant in a solution (example: for what values of
m is y = emx a solution to y´´ - 5y´ + 6y = 0)
Direction field
Euler’s method
Solving a separable equation
Exponential growth, decay
Logistic equation
Taylor Series
Use substitution to find a Taylor series
Use Taylor series to find an integral or limit of a function
Sample Questions (Caution: this is only a sample! Not a complete list of the types of
questions that will be on the test)
4
2
∑i
1) Compute
i =1
4
2) Compute
∫
3) Estimate
∫
4) Evaluate
∫
0
1
0
t 2 dt by using the Reimann Sum (limit as n approaches infinity)
e − x dx by using Simpson’s Rule. Let n = 6.
8
1
2
2 x 1 + x 2 dx
π 
cos 
 x dx
5) Evaluate ∫
x2
1
6) Evaluate ∫ 2
dx
x −1
9 − x2
7) Evaluate ∫
dx
x2
∞ ln x
8) Evaluate ∫
dx
1
x2
9) A region is bounded by the curve y = 4 – x2 and y = -x + 2. Then this region is rotated about the xaxis. Find the volume of the resulting solid.
10) A graph of f(t) is shown below.
y
2
1.5
1
0.5
x
0
2
4
6
8
-0.5
-1
a) Sketch a graph of the area function g(x) =
∫
x
1
f (t )dt
b) For what values of x is g(x) concave up (approximately)?
11) For what nonzero values of k does the function y = sin(kt) satisfy the differential equation y′′ + 9y
=0?
dx
12) Solve the differential equation xe− t
= t , with the initial condition x(0) = 1.
dt
13) Explain the first Fundamental Theorem of Calculus, and the reasoning behind it.
14) Let f(x) = e(-x^2). Find the first 6 terms of the Taylor series, by using substitution.Then use the
series to compute
1
∫e
0
− x2
dx