1. [9]
2. [15]
3. [10]
Ma 121
Name:
4. [8]
5. [8]
Exam 2
Total: [50]
Oct 11, 2012
Solutions
Check your Section:
A - Gilman (9:00am)
B - Serbin (9:00am)
C - Serbin (10:00am)
D - Gilman (10:00am)
E - Serbin (noon)
F - Miller (noon)
G - Macdonald (10:
1. [8]
2. [8]
3. [8]
Ma 121
Name:
4. [8]
5. [6]
6. [6]
7. [6]
Final Exam
Total: [50]
Oct 18, 2012
Solutions
Check your Section:
A - Gilman (9:00am)
B - Serbin (9:00am)
C - Serbin (10:00am)
D - Gilman (10:00am)
E - Serbin (noon)
F - Miller (noon)
G
Lecture 10: Linear approximations and dierentials
MA 121, Fall 2012
Lecture 10: Linear approximations and dierentials
Linear approximations
Suppose a function f (x) is given and we can compute the value of f at
x = x0 , but it is hard to compute the value
Lecture 11: Vertical and horizontal asymptotes
MA 121, Fall 2012
Lecture 11: Vertical and horizontal asymptotes
Innite limits
Denition
We write
lim f (x) =
xa
if the values of f (x) can be made arbitrarily large by choosing the values of x
close enough t
Lecture 12: Local optima. Optima on closed intervals
MA 121, Fall 2012
Lecture 12: Local optima. Optima on closed intervals
Global extrema
Denition
1
A function f (x) has an absolute (or global) maximum at x = c if
f (c) f (x) for every x Dom(f ). f (c) i
Lecture 13: Derivatives and the shapes of curves
MA 121, Fall 2012
Lecture 13: Derivatives and the shapes of curves
Increasing / decreasing functions
Denition
f (x) is increasing on an interval if f (x1 ) < f (x2 ) for any points of the interval
x1 , x2 s
Lecture 14: Optimization
MA 121, Fall 2012
Lecture 14: Optimization
Optimization
Here are some steps for solving optimization problems:
Spend some time understanding the problem: What quantity is to be
optimized? What is the independent variable? if the f
Ma 121 - Dierential Calculus
Hw-1 Cover Sheet
Name (Printed):
Due Tuesday, Sept 4, 2012
Lecture Section:
Pledge and Sign:
Solutions are to be written up on a separate piece of paper, rather than directly on this cover sheet (unless
explicitly allowed). At
Ma 121 - Dierential Calculus
Hw-5 (Cover Sheet)
Name (Printed):
Due Tuesday, Oct 16, 2012
Lecture Section:
Pledge and Sign:
Solutions are to be written up on a separate piece of paper, rather than directly on this cover sheet (unless
explicitly allowed).
Lecture 9: Derivatives of logarithms and inverse trigonometric
functions
MA 121, Fall 2012
Lecture 9: Derivatives of logarithms and inverse trigonometric functions
Inverse of sin x
Recall that a function f (x) is inverse to the function g(x) if
f (g(x) =
Lecture 6: Derivatives of trigonometric functions
MA 121, Fall 2012
Lecture 6: Derivatives of trigonometric functions
Derivatives of sin x and cos x
Consider f (x) = sin x. Let us compute f (x).
By denition we have
f (x) = lim
h0
sin(x + h) sin x
f (x + h
Ma 121 - Dierential Calculus
Hw-2 (Cover Sheet)
Name (Printed):
Due Tuesday, Sept 11, 2012
Lecture Section:
Pledge and Sign:
Solutions are to be written up on a separate piece of paper, rather than directly on this cover sheet (unless
explicitly allowed).
Ma 121 - Dierential Calculus
Hw-4 (Solutions)
Due Tuesday, Fri. 5, 2012
Legibility, organization of the solution, and clearly stated reasoning where appropriate are all
important. Points will be deducted for sloppy work or insucient explanations.
1. A tra
Lecture 4: Derivatives of polynomials and exponential functions
MA 121, Fall 2012
Lecture 4: Derivatives of polynomials and exponential functions
Power function
From the denition of the derivative it is easy to see that
d
(c) = 0
dx
for every constant c R
Lecture 3: The derivative as a function.
MA 121, Fall 2012
Lecture 3: The derivative as a function.
Denition of the derivative
Denition
The derivative of a function f (x) at a R, denoted f (a) is the value of the
limit
f (x) f (a)
f (a) = lim
xa
xa
if the
Lecture 2: The limit of a function. Limit techniques.
MA 121, Fall 2012
Lecture 2: The limit of a function. Limit techniques.
Denition of limit
Denition
The limit of a function f (x), as x approaches a R, is L R, and we write
lim f (x) = L
xa
if the value
Lecture 1: Velocity and tangents. Derivatives and rates of
change.
MA 121, Fall 2012
Lecture 1: Velocity and tangents. Derivatives and rates of change.
The Velocity Problem
A particle is moving in a straight line.
t is the time that has passed from the st
Lecture 5: The Product and Quotient Rules
MA 121, Fall 2012
Lecture 5: The Product and Quotient Rules
The product and quotient rules
Fact
If f and g are both dierentiable, then
d
d
d
[f (x) g(x)] =
f (x) g(x) + f (x)
g(x)
dx
dx
dx
Example
Dierentiate
f (
Lecture 7: The Chain Rule
MA 121, Fall 2012
Lecture 7: The Chain Rule
The Chain Rule
Fact
If g is dierentiable at x and f is dierentiable at g(x), then the composite
function F = f g dened by F (x) = f (g(x) is dierentiable at x and F is
given by the prod
Lecture 8: Implicit dierentiation
MA 121, Fall 2012
Lecture 8: Implicit dierentiation
Implicit dierentiation
An equation F (x, y) = 0 denes a function y = f (x) implicitly near some point
(x0 , y0 ) if F (x, f (x) = 0 is an identity which holds for every
Ma 121 - Dierential Calculus
Hw-3 (Solutions)
Due Tuesday, Sept 25, 2012
Legibility, organization of the solution, and clearly stated reasoning where appropriate are all
important. Points will be deducted for sloppy work or insucient explanations.
1. Supp