6.5 and Exam Practice

This week we are doing a practice test for our Applications of Integrals exam.
I am also assigning a video on 6.5: Average Value of a Function and the MVT for Integrals, since we never really formally covered this lesson.

Please watch the video below and then work through the Integration Applications Practice Exam. We will go over the practice exam and any questions on Thursday.

Blessings,

Mrs. Seekamp

Ave Value of a functions

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Discussion Board

discussion board

This Week’s Lessons: 3.7 and 3.8

This week we are covering the last of the Derivative Rules!!! We’ll be working through lessons 3.7 and 3.8.

Please show me your notes on the following two lessons this Friday.

Lesson 3.7 Assignment

• Watch Video: Derivatives of Exponential Functions
• Watch Video: Derivatives of Inverse Functions
• Watch Video: Derivatives of Logarithmic Functions

Lesson 3-8 Assignment

• Watch Video: Derivatives of Inverse Trig Functions
• Try practice problems:
  1. Find the derivative of y = arcsin (3x)
  2. Find the derivative of y = arctan (1/x)

Final/ AP Review Material

Review Documents

2008 Free-Response Questions

2008 Free-Response Solutions

2007 Free-Response Questions

2007 Free-Response Solutions

Calculus Review Sheet

Topics for Final

• Limits
  • Basic limit properties
  • Limits at infinity
  • “Mental Limits”
• Particle Motion
  • Position, velocity, acceleration
  • When is particle furthest to left/right?
  • When is particle moving forward/backward?
  • When is particle speeding up/slowing down?
• Graphical relationship between f, f ‘ and f ‘ ‘
  • Absolute Max/Min
  • Find intervals of:
    • Increase/Decrease
    • Concave up/ concave down
• Implicit Differentiation
• FTC, Parts 1 & 2, including the Net Change Theorem
• Area/Volume Problem
  • Area between two curves
  • Volume using washers or disks
  • Volume of known cross-sections
• Differential Equation
  • Solve a separable DE with an initial condition
  • Sketch/ identify a slope field

DE Exam Review

Students,

Here are review materials for your Tuesday test on Differential Equations:

Differential Equations Exam Review

Slopefields Worksheet

Happy Studying!

Ms. Seekamp

4.3 – 4.6 Lessons

Lesson 4.3 – Area

Lesson 4.4 – The Definite Integral

Lesson 4.5 – The FTC, Pt. 1

AP Problems involving the FTC, Pt. 1 (worked in class)

Lesson 4.5 – The FTC, Pt 2

Lesson 4.6: Substitution

AP Set 2: 2007 #4, #6

Please turn in the AP Set in class on Friday, February 13th.

Click here to download as a pdf.

- Ms. Seekamp

So WordPress has this new polling feature…

Notes Test Review 10-10-08

definition of continuity

(1) limit as x approaches “a” must exist

(2) the function value exists

(3) the limit is equal to the function -     lim x -> a = f(a)

If an individual limit exists you can multiply it but if not then you have to go back to does the right equal the left

chapter notes

1.1

be able to take slopes of secant lines and estimate the slope of tangent lines from that

ex. 1.1

y = x ^2 + 1 find the equation of the tan. line to f (x) at x=1

pt (1,2)

second point varies because of different lines. So keep bringing the line closer to the original point.

slope of sec. = [x^2 +1] – 2 divided by [x-1] -> slope = 2

slope of tan -> y-2 = 2 (x-1)      

1.2

definition of limit

a limit exists at “a” if the limit as x approaches “a” from the left equals the limit as x approaches “a” from the right.

Notes Of Lesson 1.5 Over Limits Involving Assymptotes and Infinity

Definition of an Asymptote: a Line

-Vertical Assymptotes (V.A.): A vertical line where the function is undefined

     Example: Graph on calculator 1/(x-2), the V.A. x=2

- Horizontal Assymptotes: A horizontal line that the func. approaches, the limit as x→+/-∞

     *The function. can go thru the H.A., the point is to find the limiting value of the function as x gets very large.

     Example: f(x)=1/x (graph on calc. to see the function) lim x→∞=0, lim x→-∞=0. So there is a H.A. at y = 0.

     lim x→0 = DNE (-/+∞). There is a V.A. at x = 0.

H.A.: the line the function approaches as x goes to +/-∞ ( if you follow the function with your finger as it approaches the horizontal line, that is your H.A.)

-If the function is a Rational function (a polynomial over a polynomial), then if the denominator can’t factor there is an infinite discontinuity.

    Example. lim x→-2 f(x)=(x+1)/(x-3)(x+2) →the denominator approaches 0. Since we can’t cancel any factors there are two infinite discontinuities.

x→-2‾ (x+1)/(x-3)(x+2) →-∞

x→-2+ (x+1)/(x-3)(x+2) → +∞

So, from these choose a value close to the right or left of the value that x is approaching, as the case may be, and plug it in to find the sign of each factor, then use that to determine the signs of the whole function to determine if it is approaching the – or + ∞, which is the limit.

 

-Finding H.A.

-find the lim as x→∞

Example: f(x )= 2-(1/x)

lim x→∞f(x)= 2- (1/x approaches 0 as the denominator gets closer to ∞)

so, lim =2 (Which is the H.A)

*****Rule******

lim x→-/+∞ 1/x^n = 0

 

Example: lim x→∞ (5x-7)/(4x+3)= +∞/+∞ ≠1 → infinity divided by infinity is an indeterminate form

* divide each term by x (the largest exponent of x in the denominator), so the problem becomes limx→∞[(5-7/x)/ (4+3/x)]

- so the n/x fraction approaches 0

-so 5/4 is the H.A. ( where x→∞)

 

So, when x approaches ∞ with 2 polymonials divided by each other, as long as the power of the highest term of each is the same, the H.A. is the coefficients of the highest terms

Example: lim x→∞ (3x^2 +7x +1)/(2x^2 + 3) → multiply by 1/x^2 → (3+ 7/x +1/x^2)/(2+3/x^2)

so, lim x→ ∞ = 3/2

 

***************Rule**************

the mathematical statement of the above can be found in theorem 5.2 on page 113

 

Examples 5.12 in the book is good example on page 117