The University of British Columbia
Midterm #1 - October 14, 2011
MATH 104
Closed book examination
Last Name:
Time: 50 minutes
First:
Student Number:
Signature:
Section Number:
Special Instructions:
No memory aids or communication devices are allowed. No c
The Fundamental Theorem of Calculus
Suppose f is continuous on [ a, b].
x
1. If g ( x) f (t ) dt , then g ( x) f ( x).
a
2.
b
a
f ( x) dx F (b) F (a ), where F is any antiderivative of f .
What is the derivative of the function g ( x )
x
0
1 t 2 dt ?
d x
Calculus 12 Quiz: Curve Sketching Name:
Graphing calculators are not pennitted.
1. Determine all local maximum and local minimum points, and am; absolute maximum or
minimum values of x) 241933. on the interval [1 I 1).
Check endPOinls 2 l3(-i) = Li(i)3 ~3
Continuity
A function f is continuous at a number a if lim x f (a) .
xa
There are three types of discontinuity to consider:
Removable discontinuities occur at "holes". They are removable
because the function could be redefined at that point, thereby
remov
Curve Sketching
How can f(x), f'(x) and f'(x) help us to sketch the graph of f(x)?
If we factor f(x), we can determine
If we factor f'(x), we can determine
If we factor f'(x), we can determine
Example:
f(x) = x3 - 3x2
Example:
Try:
f(x) = x4 - 4x3
f(x) =
Implicit Differentiation
When it is not easy (or possible) to isolate y from the rest of the
equation, we can determine the derivative using implicit differentiation.
Consider the circle x2 + y2 = 25. We will find its derivative using two
methods:
1. Isol
Introduction to Limits
The expression lim f ( x) L is read as: "the limit of f(x) as x approaches a is L".
xa
f(a) does not need to be defined in order for lim f ( x) to exist because we are
x a
concerned with the value of f(x) when x is sufficiently clos
Area between curves
How can the area between two curves be calculated? For example, let's calculate
the area of the region bounded by the parabolas y = x2 and y = 2x - x2.
1. What does this region look like?
2. What are the points of intersection?
3. How
Continuous Compound Interest
MATH 104
Mark Mac Lean
2013W
Recall from your high school studies that the compound interest formula
is
r nt
,
n
where P is the principal, r is the annual interest rate as a fraction, n is the
number of compounding periods per
Volume - Spinning in all directions
How much material is required to make an object? Let's examine Titus' chew toy:
Bounded regions can be rotated about lines other than the x-axis.
Consider the region bounded
by y = x and y = x2. What does
it look like?
Area between curves, continued
Find the area of the region bounded by the line y = x - 1 and the
parabola y2 = 2x + 6.
y
x
Let's look at this problem from a different angle.
Let's look at this problem from a different angle.
y
y
y
x
x
x
More Related Rates
Example 3: A water tank has the shape of an inverted circular cone with base
radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2
m3/min, find the rate at which the water level is rising when the water is 3 m
Differential Equations
A differential equation involves an unknown function and its derivative.
dy
ky
dt
Can you think of a function whose derivative is a constant multiple of itself?
We can show that this is the only type of solution of this differentia
The First Derivative Test
A function that is only increasing on an interval I, or only decreasing on an
interval I, is said to be monotonic on I.
If we want to find out where a function is increasing, or where it is decreasing,
we can apply the First Deri
Separable Differential Equations
Some examples of other differential functions are:
y xy
y 2 y y 0
y xy y 22 y e x
A separable equation is a first-order differential equation that can be written in
the form
dy
g ( x) f ( y )
dx
since it can be rearranged
Trigonometric Functions
Limits as 0 of trigonometric functions:
lim sin
lim cos
0
0
lim tan
0
Fundamental Trigonometric Limit:
lim
0
Determine lim
0
cos 1
sin
1
.
sin 7
and lim cot .
0
0
4
Calculate lim
Derivatives of trigonometric functions:
d
The Area Problem
How can we calculate the area between the curve of a function and the x-axis?
For example, consider the function f(x) = x2 + 1, and suppose we want to find the
area under the curve (and above the x-axis) on the interval [0, 2]. We will lo
A tank contains 20 kg of salt dissolved in 5000 L of water. Brine that contains
0.03 kg of salt per liter of water enters the tank at a rate of 25L/min. The solution
is kept thoroughly mixed and drains from the tank at the same rate. How much
salt remains
Optimization: Calculus Style!
You have most likely seen these examples before and used a graphing calculator
to solve them. You can put that graphing calculator away now! We're going to
solve these problems analytically.
Example 1: An open-box top is to b
Related Rates: Basic Questions
Example 1: Air is being pumped into a spherical balloon so that its volume
increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing
when the diameter is 50 cm?
Start by identifying the given informa
Neilinder Saini
Assignment Assignment1 due 09/17/2014 at 08:00am PDT
1. (1 pt)
Evaluate the limit, by using the appropriate Limit Law(s).
lim (3x4 + 2x2 x + 4)
MATH104-184-ALL 2014W1
6. (1 pt)
What is wrong with the following equation?
x2
x2 + x 6
= x+3
x