1. Calculus AB Topics Limits Continuity, Asymptotes

2. Do not give a Precalculus answer on a Calculus exam.
Consider
Is there a vertical asymptote at x = 3?
Yes, the right-hand limit = ∞
Therefore x = 3 is an asymptote
Do not give a Precalculus answer on a Calculus exam.

3. Consider Is there a horizontal asymptote at y = 0?
Yes, the limit as x goes to ∞ is 0
Therefore y = 0 is an asymptote

4. Definition of Continuity
Continuity at an interior point –
Continuity at an endpoint -

5. Derivative Highlights Limit Definitions of Derivative

6. Do you REALLY know the definitions of a derivative???
What are they???

7. Example

8. What does it mean for a function to be differentiable?
The derivative exists at all x values or at a specified x value
For a derivative to exist…the LEFT hand derivative, must equal the RIGHT hand derivative.
What is the example of a function that is continuous at a specific x value but not differentiable at that x-value?

9. quotient rule: product rule:
Notice that this is not just the product of two derivatives.
quotient rule:

11. Summary of Trig Derivatives
GOLDEN TICKET!!!

12. Definition of the Chain Rule
Derivative of
“outside”
function
Derivative of
“inside”
function

13. Implicit Differentiation
Derive “in place”
Power rule
Chain rule
Derivative of
Constant is 0

14. Inverse Trig Rules
GOLDEN TICKET!!!

15. Exponentials and Logs
GOLDEN TICKET!!!

16. Applications of Derivatives

17. Extrema and Concavity
Critical points are where the derivative is zero or undefined.
Extrema may occur at critical points and endpoints
Finding extrema on a closed interval—TEST the ENDPOINTS (Extreme Value Theorem)
1st derivative tells you where the function is increasing/decreasing and possible extrema
2nd derivative tells you where the function is concave up/down and possible points of inflection.

18. Sign charts
Sign charts are valuable tools and are allowed, BUT THEY ARE NEVER NEVER NEVER SUFFICIENT TO EARN A POINT
To earn the test points you must interpret the sign chart using words
Your words should demonstrate that you understand the connection between the positive/negative behavior of a graph and the increasing/decreasing/extrema behavior of the parent function and how the increasing/decreasing behavior determines the extrema behavior

19. Intermediate Value Theorem
If a function is continuous on [a,b] then the function takes on all values between f(a) and f(b).

20. Mean Value Theorm
If f is continuous on [a,b] and differentiable on (a,b)
Then there exists a number, c, in (a,b) such that

21. Optimization
Write one or two equations that model the situation described.
You may need to write two equations and use substitution.
Take the derivative to find the maximum or minimum.

22. 1. An open top box is to be made by cutting congruent squares from the corners of a 20-by-25 inch sheet of tin and bending up the sides. What dimensions give the box the LARGEST volume? (ex 1 – pg. 206)
Steps:
Set up an equation to maximize or minimize
1a. You may have 2 equations and substitute into one.
2. Take its derivative.
Set derivative = 0 and check slopes to show it is a max/min.
Answer the question asked.

23. Linearization A linearization is just another term for tangent line!
You can use a linearization to estimate the value of a function at a given x-value.
The closer the x-value is to the point of tangency the better the estimation.

24. Related Rates Draw a picture to model the problem
Write an equation that reflects the model
Pythagorean Theorem
Trig
Similar Figure (such as cones)
Plug in values that never change
Implicit differentiation
Plug in values that do change and solve

25. Integration

26. Riemann Sums
LRAM
RRAM
Midpoint
- midpoint of x-interval, not the geometric midpoint
Trapezoidal
If intervals are not uniform width, rectangles and trapezoids must be calculated individually

27. Definition of Integral

28. Fundamental Theorem of Calculus

29. FTC – Evaluative Part

30. Average Value Theorem: If a function is integrable on [a,b], then its average y-value is:

31. Differential Equations

32. Is called a definite integral.
We can evaluate it and get a numerical answer.
Is called an indefinite integral.
Its solution is the set of all possible antiderivatives.

33. Finding “c” Solve with the initial condition F(1) = 4
1. Find the antiderivative family
2. Substitute the given initial condition values
3. Solve for c
4. Substitute back into antiderivative

34. Compare and Contrast Solve with the initial condition F(1) = 4
This is called a differential equation. You solve it the same way.
Your solution is y = equation.

35. A slopefield is a tool to learn the characteristics of the solution to an indefinite integral without actually knowing the antiderivative.
You must be able to
Draw a slopefield
Indicate a particular solution on a slopefield
Answer questions about a slopefield
Graphed solutions:
Include the given point
Do not cross discontinuities
Do not cross slope lines

36. Separable Differential Equations
Separate the x’s and y’s
Integrate each side
Solve for c if you can
Solve for y

37. Diff Eq Review
You’ve deposited $1000 in Calculus National Bank. The bank has promised your money will grow at the rate
Solve the differential equation.

38. uSubstitution:

39. Inverse Trig Integrals
What inverse trig function looks like the best fit?
Get the 1 in the bottom
Factor constants outside the integral
Write as (something)2
Use u-Substitution with inverse trig formula

40. Applications of Integrals
Our Last Unit!!

41. Velocity is the derivative of position.
Position is the integral of velocity.
Displacement is the distance from an arbitrary starting point at the end of some interval. It is the difference between ending position and starting position.
Total distance is the how far an object travelled regardless of direction.
Position
Velocity
Acceleration
Derivatives
Integrals

42. Summary To find You Displacement Total Distance
Position at a specific time
Solve the indefinite integral and use given initial condition to find “c”

43. Other Applications Velocities (distance and displacement)
Rates of flow (e.g. oil leaking from a tanker or water flowing into a container)
How many cars flow through an intersection
How much money has accumulated in a bank account
Anytime you know the rate of something and what to know how something has accumulated.

44. What if we want area between two curves?

45. Why does x always have all the fun?b
Let’s integrate with respect to y a b

46. Known Cross-sections

47. Disks and Washers
Disk method
Washer method

48. The axis of rotation is important
Same equations = same area being rotated.
But different axis of rotation = different solid