2.5: Continuity
When we calculate the limit of a function as x approaches a value a, we ignore the behavior of
the function at x = a for technical reasons. Yet in certain contexts, we do actually care about
the behavior of the function at a. We might want
2.2: The Limit of a Function
Motivation/ background: the tangent line problem (of which the velocity problem is a special case) is one of the main motivating problems for the concept of limit, which we will
define shortly. We defined the slope of the tang
2.8: The Derivative as a Function
Summary of lesson: In the last class, we stated the limit definition for the slope of the tangent line to a curve at a point (a, f (a). Another name for this slope is the derivative. Today
we extend this definition to con
3.10: Linear Approximations and Differentials
Summary of lesson: Computing the outputs of complicated functions can be expensive for
computers to do. In another scenario, solving a system of equations that involves functions
that are not nice is a difficu
3.5: Implicit Differentiation
Summary of lesson: So far we have found derivatives of functions in which a dependent
variable y is expressed explicitly in terms of an independent variable x. Sometimes the
relationship between x and y is expressed more natu
Calc 1A: Exam 3 Review and Additional Problems
Exam 3 will take place the first half of class on Wednesday, May 3. If you suspect youll need
extra time, you may come in at 5:30 to get started early.
The exam will address material from sections 3.10 and 4.
3.4: The Chain Rule
Summary of lesson: Composite functions appear quite frequently in math, yet we have not
yet developed a shortcut rule that enables us to differentiate any such function. The chain
rule is the missing link.
Lets say you work for a choco
Math 1A
Name:
Quiz 1
Due: Wed, 2/22/2017
Write your solutions on separate sheets of paper.
Please show your work and/or explain your answers!
You may work with other students on the quiz, but you should understand
your own write-up. ,
1. (5 points) Sketch
Section 3.6
Derivatives of Logarithmic Functions
In this section, we use implicit differentiation to find the derivatives of the
logarithmic functions = . Logarithmic functions are
differentiable.
Can you guess why?
A.
Proof:
Calc 1A: Exam 2 Review and Additional Problems
Exam 1 will take place the first half of class on Wednesday, March 29. If you suspect youll
need extra time, you may come in at 5:30 to get started early.
The exam will address material from sections 3.1-3.7
2.7: Derivatives and Rates of Change
Summary of lesson: We return to one of the motivating applications of limits: finding the
slope of a tangent line to a curve. In the case of a position function, the slope of a tangent line
(which is obtained by taking
2.3: Calculating Limits Using the Limit Laws
Graphs and tables of values are not always sufficient for precisely evaluating limits. To cite
a couple of scenarios, the value of a limit might not be a nice number, in which case its
hard to pinpoint from a g
Final Exam Review
The final exam, which is cumulative, is on Monday, May 22, from 6-8 pm.
You may bring one 8.5 by 11 in sheet of paper filled on both sides with
information of your choice. You may also use a scientific calculator.
For the final, I will c
4.3: How Derivatives Affect the Shape of a Graph
Summary of lesson: The first and second derivatives provide important information about
the shape of a functions graph. The first derivative tells us when the function is increasing
or decreasing, and the s
Unit 3 (Cool Applications of the Derivative) HW
*Note: due dates are tentative and may be later than written. In particular, if we get behind
schedule, a section originally due one week will be pushed back to be due the next week. Also,
even though I will
4.2: The Mean Value Theorem
Summary of lesson:
Any real-valued differentiable function that attains
equal values at two distinct x-values must have a point between them where
. More generally, the average rate of change of
a differentiable function on a c
3.2: The Product and Quotient Rules
Summary of lesson: Many real-world functions are expressed as products or quotients
of functions. (For an example, see Problem 5 in these notes.) To find rates of change of
such functions, it is helpful to have addition
Unit 4 (The Derivatives Unifying Cousin: the Integral) HW
*Note: due dates are tentative and may be later than written. In particular, if we get behind
schedule, a section originally due one week will be pushed back to be due the next week. Also,
even tho
2.6: Limits at Infinity; Horizontal Asymptotes
The limits lim f (x) we have discussed so far are not the only types. For those, we wanted
xa
to examine the behavior of a function as the independent variable (eg. time) approached a
finite value. In other s
Unit 1 (Know Your Limits: An Intro to Calculus) HW
*Note: due dates are tentative and may be later than written. In particular, if we get behind
schedule, a section originally due one week will be pushed back to be due the next week. Also,
even though I w
Math 1A
Name:
Quiz 2
Due: Wed, 3/29/2017
Write your solutions on separate sheets of paper.
Please show your work and/or explain your answers!
You may work with other students on the quiz, but you should understand
your own write-up. ,
1. (3 points each) C
Unit 2 (Derivatives: The World is a Changing) HW
*Note: due dates are tentative and may be later than written. In particular, if we get behind
schedule, a section originally due one week will be pushed back to be due the next week. Also,
even though I wil
3.9: Related Rates
Summary of lesson: In problems involving several related quantities that are changing
simultaneously, we might want to know: how does changing one of the quantities affect each
of the others?
Problem-Solving Guidelines:
1. Read the prob
3.3: Derivatives of Trigonometric Functions
Summary of lesson: Trigonometric functions show up throughout the sciences, in particular
any time a variable quantity has periodic or wavelike behavior. Hence, we turn our attention
to studying their derivative
3.1: Derivatives of Polynomials and Exponential Functions
Summary of lesson: Computing derivatives longhand using the limit definition can be timeconsuming, tedious, and sometimes difficult. Fortunately, there are shortcut rules for finding
derivatives. I
Calc 1A: Exam 1 Review and Additional Problems
Exam 1 will take place the first half of class on Monday, March 6. If you suspect youll need
extra time, you may come in at 5:30 to get started early.
The exam will address material from chapter 2. (Section 2
Section 3.7
Rates of Change in the Natural and Social Sciences
CD
We know that if = (), then the derivative can be interpreted as the rate of change
CE
of with respect to . In this section, we examine some of the applications of this idea.
Not
4.1: Maximum and Minimum Values
Summary of lesson: One of the big applications of calculus (namely, the derivative) is the
ability to find maximum and minimum values of functions.
Key Question: Given a function y = f (x), how can we figure out the maximum
4.4: Indeterminate Forms and LHospitals Rule
Summary of lesson: When two functions go to 0 or simultaneously and we want to
know which gets small or big faster, we can sometimes use algebraic arguments to settle
matters. At other times algebra will do us