Sample Midterm 1:
Q1:[5 marks] a) State the denition of the derivative.
b) Using the denition, nd the derivative of f (x) = x2 + x.
Q2: [1 mark] Fill in the blanks: A function is continuous at x = a if (i)
lim f (x)
; (ii) f (a)
; and (iii) f (a) =
Q3: [4
Class outline for class 5
July 16, 2014
Text sections:
3.3: Derivatives of trigonometric functions
3.4: The chain rule
3.5: Implicit Dierentiation
Theory:
Find the derivative of trigonometric functions (memorize!).
Determine limits of trigonometric f
Class outline for class 6
July 21, 2014
Text sections:
3.6: Derivatives of Logarithmic Functions
3.7: Rates of Change in the Natural and Social Sciences
3.8: Exponential Growth and Decay
Theory:
What is logarithmic dierentiation?
Understand how the d
Class outline for class 6
July 22, 2013
Text sections:
3.7: Rates of Change in the Natural and Social Sciences
3.8: Exponential Growth and Decay
3.9: Related Rates
3.10: Linear Approximations and Dierentials
Theory:
Understand how the derivative can
Class outline for class 7
July 23, 2014
Text sections:
3.9: Related Rates
3.10: Linear Approximations and Dierentials
Theory:
What is a linear approximation?
What is a dierential ?
Calculate:
Solve related rates problems.
Find the linear approximati
Class outline for class 7
July 24, 2013
Text sections:
4.1 Maximum and Minimum Values
4.2 The Mean Value Theorem
4.3 How Derivatives Aect the Shape of the Graph
Theory:
What is an absolute maximum? Absolute minimum?
What is a local maximum? Local min
Class outline for class 8
July 29, 2013
Text sections:
4.4 Indeterminate Forms and LHospitals Rule
4.5: Summary of Curve Sketching
Theory:
What is an indeterminate form?
What is LHospitals Rule?
What is an even function? What is an odd function?
Wha
Class outline for class 8
July 28, 2014
Text sections:
4.1 Maximum and Minimum Values
4.2 The Mean Value Theorem
4.3 How Derivatives Aect the Shape of the Graph
Theory:
What is an absolute maximum? Absolute minimum?
What is a local maximum? Local min
Class outline for class 9
July 31, 2013
Text sections:
4.4 Indeterminate Forms and LHospitals Rule
4.5: Summary of Curve Sketching
Theory:
What is an indeterminate form?
What is LHospitals Rule?
What is an even function? What is an odd function?
Wha
Class outline for class 9
July 31, 2013
Text sections:
4.7: Optimization Problems
4.8: Newtons Method
4.9: Antiderivatives
Theory:
What is Newtons Method?
What is an antiderivative?
Calculate:
Solve optimization problems.
Apply Newtons method to ap
Class outline for class 10
August 6, 2014
Text sections:
4.7: Optimization Problems
4.8: Newtons Method
Theory:
What is Newtons Method?
Calculate:
Solve optimization problems.
Apply Newtons method to approximate values.
1
Optimization Problems
A tra
Class outline for class 11
August 12
Text sections:
5.3 The Fundamental Theorem of Calculus
5.4 Indenite Integrals and the Net Change Theorem
5.5 Substitution Rule
Theory:
What is the Fundamental Theorem of Calculus?
What is an indenite integral?
Wh
Class outline for class 10
August 7, 2013
Text sections:
5.1: Areas and Distances
5.2 Denite Integrals
Theory:
What is area?
What is distance?
What is a Riemann sum?
What is an integrand?
What are limits of integration?
What is the Midpoint Rule?
Class outline for class 12
August 13, 2014
Text sections:
5.3 The Fundamental Theorem of Calculus
5.4 Indenite Integrals and the Net Change Theorem
Theory:
What is the Fundamental Theorem of Calculus?
What is an indenite integral?
What is the Net Cha
Class outline for class 5
July 17, 2013
Text sections:
3.4: The chain rule
3.5: Implicit Dierentiation
3.6: Derivatives of Logarithmic Functions
Theory:
What is the chain rule?
What is implicit dierentiation?
Find the derivative of inverse trigonome
Class outline for class 3
July 9, 2014
Text sections:
2.7: Derivatives and Rates of Change
2.8: The Derivative as a Function
1.5: Exponential Functions
Theory:
What is the derivative of a function at a number?
What is an average rate of change?
What
Class outline for class 4
July 14, 2014
Text sections:
1.6: Inverse Functions and Logarithms
3.1: Derivatives of Polynomials and Exponential Functions
3.2: The product and quotient rules
Theory:
What is a one-to-one function?
What is the horizontal l
Math 1000 Course Notes
Rob Noble
November 28, 2014
2
Contents
2 Limits and derivatives
2.1 The tangent and velocity problems . . .
2.2 The limit of a function . . . . . . . . . .
2.3 Calculating limits using the limit laws .
2.5 Continuity . . . . . . . .
Initial Value Differential Equations and Applications to other areas of Science
Theory 1: From the Chain rule we have:
(1)
If g(x) = ln(f (x) + C then
f (x)
g (x) =
f (x)
(2)
If g(x) = (f (x)n + C then
g (x) = n(f (x)n1 f (x)
and
Reversing the procedures
Sample Test 2 Partial Solutions
Page 215 Q6:
p
p
2 x+ y
2 1/2 1 1/2 0
x
+ y
y
2
2
y0
= 3
= 0
=
p
2y 1/2
= 2 y/x
x1/2
Page 215, Q 22
Find g 0 (0) if
g(x) + x sin(g(x) = x2
Note that at x = 0
g(0) + 0 sin(g(0) = 02 ! g(0) = 0
Dierentiating
0
g (x) + sin(g(x
Sample Midterm 1:
Q1:[5 marks] a) State the denition of the derivative.
b) Using the denition, nd the derivative of f (x) = x2 + x.
Q2: [1 mark] Fill in the blanks: A function is continuous at x = a if (i)
lim f (x)
; (ii) f (a)
; and (iii) f (a) =
Q3: [4
f ( x, t ), a < x vt < b
D( x, t ) =
0, others
2
2
v
D( x, t ) = A sin( kx t + 0 ) , k =
, = 2f =
= kv , = vT =
T
f
n=
1.
:
Sinusoidal Waves:
String Waves:
v=
c vac
=
v mat
(1) Pulse Wave:
Ts
D ( x, t ) = a sin( kx + t ) + a sin( kx t ) = 2a sin( kx )
Name:
Math 112b Final Exam
May 4, 2010
Your signature:
Math 112b Section:
Instructions: There are ten problems, with the number of points for each problem indicated at the start of the problem. There is a total of 250 points. You
are not allowed the use o
Class outline for class 2
July 8, 2013
Text sections:
2.3: Calculating Limits using the Limit Laws
2.5: Continuity
2.6: Limits at Innity; Horizontal Asymptotes
Theory:
Properties of limits.
Section 2.3: Theorems 1, 2, 3 (Squeeze Theorem).
Section 2.
Class outline for class 2
July 7, 2014
Text sections:
2.3: Calculating Limits using the Limit Laws
2.5: Continuity
2.6: Limits at Innity; Horizontal Asymptotes
Theory:
Properties of limits.
Section 2.3: Theorems 1, 2, 3 (Squeeze Theorem).
Section 2.
Class outline for class 1
July 3, 2013
MATH 1000
Is this the course for you?
MATH 1215, MATH 1500
MATH 1000X/Y
MATH 1000 (Fall/Winter of 2013/2014)
Do you have a strong enough foundation in
mathematics for this course?
diagnostic test
Calculus: Fear N
Class outline for class 3
July 10, 2013
Text sections:
2.7: Derivatives and Rates of Change
2.8: The Derivative as a Function
1.5: Exponential Functions
1.6: Inverse Functions and Logarithms
Theory:
What is the derivative of a function at a number?
Class outline for class 4
July 15, 2013
Text sections:
3.1: Derivatives of Polynomials and Exponential Functions
3.2: The product and quotient rules
3.3: Derivatives of trigonometric functions
Theory:
Rules of dierentiation:
Power Rule
The Constant
MATH 1000 Winter 2015
Time: 12:35-1:25
No books, cheat sheets or calculators are allowed. Simplification is not required. There are 5
pages and 6 questions.
Name:_
Banner Id: =
B
0
0 a
B
0
0
b
c
d
e
f
Q1[15 Marks]
(a) Let f (x) = (a + 1) sin(x) + (c + a)x
Math 1000 Winter term: Make-up Midterm
50 mins-no cheat sheets, notes, books or phones.
Name:_ Banner Id: B00
Q1[15]: differentiate the following
(i) f (x) = 5x4 + sin(x) + x 1/4 ; f 0 (x) =
(ii)
x + x1/2
f (x) = 2
; f 0 (x) =
x + cos(x)
Sample Midterm 3:
Q1:[5marks] a) State the definition of the derivative.
b)Using the definition, find the derivative of f (x) = x2 + x.
Q2:[1mark] Fill in the blanks: A function is continuous at x = a if (i) lim f (x)
;
(ii) f (a)
; and (iii) f (a) =
.
Q3