- General review:
- Date due: August 27 Tuesday.
- Reading from the textbook:
*Skim*: Everything through Section 1.2 (everything through page 18);*Skim*: Section 1.6 through "Finding Inverses" (pages 38–41).

- Reading from my notes:
- Everything through Section 1.4 (everything through page 6);
*Optional*: Section 1.5 (page 7).

- Exercises due:
- If
*f*(*x*) =*x*^{2}for all*x*and*u*= 2*x*+ 3, then what is*f*(*u*)? - If
*x*+*y*= 1 and*x*−*y*= 3, then what are*x*and*y*? - If
*y*= 3*x*+ 2, then what is*y*|_{x=4}?

- If

- Limits informally:
- Date due: August 28 Wednesday.
- Reading from the textbook:
- Section 2.2 through "An Informal Description of the Limit of a Function" (pages 58–61);
- Section 2.4 through "Limits at Endpoints of an Interval" (pages 78–80);
- Section 2.6 through
the first paragraph of "Finite Limits as
*x*→ ±∞" (page 96); - Section 2.6: "Infinite Limits" before Example 13 (pages 102&103).

- Exercises due:
- Fill in the blank:
If
*f*(*x*) can be made arbitrarily close to*L*by making*x*sufficiently close to (but still distinct from)*c*, then*L*is the _____ of*f*(*x*) as*x*approaches*c*. - Yes/No:
If
*f*(*x*) exists whenever*x*≠*c*but*f*(*c*) does not exist, then is it possible that lim_{x→c}*f*(*x*) exists? - Yes/No:
If lim
_{x→c+}*f*(*x*) and lim_{x→c−}*f*(*x*) both exist and are equal, then must lim_{x→c}*f*(*x*) also exist?

- Fill in the blank:
If

- Continuity informally:
- Date due: August 29 Thursday.
- Reading from my notes: Chapter 2 through Section 2.1 (pages 9&10).
- Reading from the textbook: Section 2.5 through "Continuity at a Point" (pages 85–88).
- Exercises due:
- If
*f*(*x*) can be made arbitrarily close to*f*(*c*) by making*x*sufficiently close to (but still distinct from)*c*, then*f*is _____ at*c*. - Suppose that
*f*(*x*) exists whenever*x*≠*c*but*f*(*c*) does not exist. Is it possible that*f*is continuous at*c*?

- If

- Defining continuity:
- Date due: September 3 Tuesday.
- Reading from my notes: Section 2.2 (pages 10&11).
- Reading from the textbook: Section 2.3: "Examples: Testing the Definition", "Finding Deltas Algebraically for Given Epsilons" (pages 70–74); note that nearly all of these examples are for continuous functions, so pretend that they're using the definition of continuity from my notes, which actually makes things slightly simpler (since you can ignore the ‹0 <› part).
- Exercises due (fill in the blank):
Suppose that
*f*is a function and suppose that*c*is a real number. For simplicity, suppose that*f*is defined everywhere.- Also suppose that,
no matter what positive real number
*ε*I give you, you can respond with a positive real number*δ*so that, no matter what real number*x*I give you, as long as |*x*−*c*| <*δ*, then |*f*(*x*) −*f*(*c*)| <*ε*. This means that*f*is _____ at*c*. - Instead suppose that
I can find a positive real number
*ε*so that, no matter what positive real number*δ*you respond with, I can find a real number*x*, such that |*x*−*c*| <*δ*but |*f*(*x*) −*f*(*c*)| ≥*ε*. This means that*f*has a _____ at*c*.

- Also suppose that,
no matter what positive real number

- Defining limits:
- Date due: September 4 Wednesday.
- Reading from my notes: Section 2.5 (pages 13&14).
- Reading from the textbook:
- Section 2.5: "Continuous Extension to a Point" (pages 93&94);
*Optional*: Section 2.3 through "Definition of Limit" (pages 69&70);*Optional*: Section 2.4: "Precise Definitions of One-Sided Limits" (pages 80 and 81);*Optional*: Section 2.6: the rest of "Finite Limits as*x*→ ±∞" through Example 1 (pages 97&98);*Optional*: Section 2.6: "Precise Definitions of Infinite Limits" (pages 104&105).

- Exercises due:
- Suppose that
*f*is a function defined everywhere except at*c*, and define a new function*g*so that*g*(*x*) =*f*(*x*) whenenver*x*≠*c*but*g*(*c*) =*L*. If*g*is continuous at*c*, then*L*is the _____ of*f*approaching*c*. - Suppose that
*f*is always positive and the limit of 1/*f*approaching*c*is 0. (That is,*f*(*x*) > 0, and lim_{x→c}(1/*f*(*x*)) = 0.) Then what is the limit of*f*approaching*c*? (That is, lim_{x→c}*f*(*x*) = _____.) - Given a function
*f*, define a new function*g*so that*g*(*t*) =*f*(1/*t*) for all possible*t*, and suppose that the limit of*g*approaching 0^{+}is*L*. (That is, lim_{t→0+}*f*(1/*t*) =*L*.) What is the limit of*f*approaching infinity? (That is, lim_{x→∞}*f*(*x*) = _____.)

- Suppose that

- Evaluating limits and checking continuity:
- Date due: September 5 Thursday.
- Reading from my notes: Sections 2.6&2.7 (pages 14–17).
- Reading from the textbook:
- The rest of Section 2.2 (pages 61–65);
- Section 2.5: "Continuous Functions", "Inverse Functions and Continuity", "Continuity of Composites of Functions" (pages 88–91);
- Section 2.3: "Using the Definition to Prove Theorems" (page 74);
- Section 2.4: "Limits Involving (sin
*θ*)/*θ*" (pages 81–83); - Section 2.6:
The rest of "Finite Limits as
*x*→ ±∞", "Limits at Infinity of Rational Functions" (pages 98&99); - Section 2.6: Examples 13&14 (pages 103&104);
- Section 2.6: "Dominant Terms" (pages 106&107).

- Reading from my notes: the very bottom of page 12 through page 14 (§2.5).
- Exercises due:
- If you're taking the limit of a rational expression
as
*x*→*c*, and you get 0/0 when you evaluate the expression at*x*=*c*, then what factor can you cancel from the numerator and denominator to simplify your expression (and then evaluate the limit)? - If you're taking the limit, as
*x*→ ∞, of a rational expression whose numerator has degree*m*and whose denominator has degree*n*, then what should you factor out of both numerator and denominator to guarantee that you can evaluate the limit by doing calculations with infinity? - What is the limit of (sin
*x*)/*x*as*x*→ 0?

- If you're taking the limit of a rational expression
as

- Theorems about continuous functions:
- Date due: September 9 Monday.
- Reading from my notes:
*Optional*: Section 2.8 (pages 17&18).- Section 2.9 (pages 18&19).

- Reading from the textbook:
- Section 2.5: "Intermediate Value Theorem for Continuous Functions" (pages 91–93);
- Section 4.1 through "Local (Relative) Extreme Values" (pages 212–215).

- Exercises due:
- For each of the following circumstances,
state whether a continuous function
*f*defined on [0, 1]*must*have a root (aka a zero, a solution to*f*(*x*) = 0) or*might not*have a root under those circumstances:*f*(0) < 0 and*f*(1) < 0,*f*(0) < 0 and*f*(1) > 0,*f*(0) > 0 and*f*(1) < 0,*f*(0) > 0 and*f*(1) > 0.

- For each of the following intervals,
state whether a continuous function defined on that interval
*must*have a maximum on the interval or*might not*have a maximum on the interval:- [0, 1],
- [0, ∞),
- (0, 1],
- (0, ∞).

- For each of the following circumstances,
state whether a continuous function

- Differences and difference quotients:
- Date due: September 10 Tuesday.
- Reading from the textbook: Section 2.1 (pages 51–56).
- Reading from my notes: Chapter 3 through Section 3.1 (pages 21&22).
- Exercises due:
Suppose that
*f*is a function, and for simplicity, assume that*f*is defined everywhere. Let*y*=*f*(*x*).- Write down a formula for Δ
*y*, using*f*,*x*, and Δ*x*. - Write down a formula for
the average rate of change of
*f*on [*a*,*b*], using*f*,*a*, and*b*. - Write down a formula for
the average rate of change of
*y*with respect to*x*, using*f*,*x*, and Δ*x*.

- Write down a formula for Δ

- Derivatives as limits:
- Date due: September 11 Wednesday.
- Reading from the textbook: Chapter 3 through Section 3.1 (pages 116–118).
- Exercises due:
Suppose that
*f*is a function and*c*is a number in the domain of*f*.- Write down a formula for
*f*′(*c*) (assuming that it exists) as a limit of an expression involving values of*f*. - If
*f*′(*c*) exists, then it is the _____ of*f*at*c*. - The line through the point (
*c*,*f*(*c*)) whose slope is*f*′(*c*) (if that exists) is _____ to the graph of*f*at that point.

- Write down a formula for

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