Suzhou March 2015Suzhou Special Topics

1、 FTC- initial plus net change- linear to calculus y=b+mx;y=initial+rate*x; y=fa+f(a)(x-a) abFxdx=Fb-Fa Fb=Fa+abFxdxInitial Conditions and displacementShow any and all set up you used for your answers.1. A particle moves along the x axis. Given that the velocity of the particle is and its initial pos

2、ition is , find the position of the particle at .2. A particle moves along the x axis. Given that the velocity of the particle is and its position at is , find the position of the particle at .3. Calculator: A particle moves along the x axis. Given that the velocity of the particle is (note:you dont

3、 know the technique for the antiderivative) and its initial position is , find the position of the particle at .4. Calculator: A particle moves along the x axis. Given that the velocity of the particle is and its position at is , find the position of the particle at .5. Traffic can be modeled by a m

4、athematical equation. Rush hour in many cities is typically 7am-9am. In Chicago, the traffic engineers want to estimate the number of cars on a freeway and the amount of tolls ($.50 per car on their particular stretch) they can collect for any one morning. Before 7am, there are 500 cars that pass th

5、rough the toll. From 7am()-9am(), the cars per minute can be modeled by .a. How many cars per minute pass through the toll at 8:30am?b. How many cars pass the toll booth before 9am?c. How much money is collected before 9am?CalculusName_Introduction to Integrals and FTCObjectives: Students will estim

6、ate the total distance that an object moves, given a velocity table and using d=rt.Part I. An object is moving along a line that never decreases its velocity. The table gives the velocity at certain intervals. T, seconds0246810Velocity in meters/second81115182325Estimate the distance the object move

7、d.Use a graph if necessary. -Discuss before we move on.Do the estimation again, but use more intervals.T, seconds012345678910Velocity in meters/second89111515171823232425Discuss your methods.Summarize the following (Teacher lead):1. Riemann sums2. Left end points3. Right end points4. Midpoints5. Tra

8、pezoidal rulePart II.We want to see what happens to a graph that is not strictly increasing and the Riemann sums that are associated with it.Graph on and find the Riemann sums involved, using ten subintervals. Draw a great graph and show how you got your answer. Be as accurate as possible.Questions:

9、1. Do Riemann sums, when dealing with velocity, always find the total distance traveled or displacement (net distance)? Explain.2. What if we used more and more information? For example, data by each .5 seconds? .25 seconds? .01 seconds? Etc.Euler Method and Riemann slides for TI NspireProperties of

10、 Definite Integrals WorksheetStudents Construct using Applets and Paper1. Write a definite integral that satisfies the properties below and the conditions given. Numerous correct answers are possible. Check with your neighbors.2. Graph and use the integral function on the grapher to check your value

11、s.3. Sketch/color your representation on the graph. (Shade the area corresponding with your integral.)Properties: abfxdx=-bafxdx; aafxdx=0; abfxdx+bcfxdx=acfxdx1. a=0, definite integral is positive.Definite Integral:2. a=0, definite integral is negative.Definite Integral:3. a=-3, definite integral i

12、s positive.Definite Integral:4. a=-3, definite integral is negative.Definite Integral:5. -3a0 and -3b0, definite integral is positive.Definite Integral:6. -3a0 and -3b0, definite integral is negative.Definite Integral:7. Definite integral = 0. Show 3 different ways with different properties used.Def

13、inite Integral:7.Definite Integral: 7.Definite Integral: 8. Show the 3rd property where abc.Definite Integral: 9. Show the 3rd property where acb.Definite Integral: Calculus TheoremsIntermediate Value TheoremA function y = f(x) that is continuous on a closed interval a, b takes on every value c betw

14、een f(a) and f(b).Mean Value Theorem If y = f(x) is continuous at every point of the closed interval a, b and is differentiable at every point of its interior (a, b), then there is at least one number c between a and b such that Rolles Theorem If y = f(x) is continuous on a, b and is differentiable

15、(a, b), and if f(a) = f(b) , then there is at least one number c between a and b such that f (c) = 0Mean Value Theorem for Definite IntegralsIf f(x) is continuous on a, b, then at some point c in a, b,Average (Mean) Value of a FunctionIf f(x) is integrable on a, b, then its average (mean) value on a

16、, b is Fundamental Theorems of CalculusIf a function f is continuous on a, b and F is an anti-derivative of f, thenSecond Fundamental Theorem of CalculusExtension of 2nd Fundamental Theorem of CalculusCross Sectional AreasName_AP Calc Section7.2In 7.2, we looked at cross sections that were circles.

17、This came about because we rotated a curve about an axis. What if we have different figures besides circles? How do we find the volume?The volume is found by summing the cross sectional areas multiplied by a small width (). If we can find the area of the cross sections, we are in business.r Circle S

18、quarer is hypotenuse r is leg r is the diam. Right Isosceles Triangles of semi-circle In all of these cases, r was found to be the radius if the function was rotated about the axis. In the square and the right Isosceles triangles, the r is just the length of a side. For the semi-circle, the r is diameter of the semi-circle.Ex. 2003 AP Free Response #1Let R be the shaded region bounded by the graphs of and and the vertical line a) Find the area of R.b) Find the volume of the solid generated when R is revolved ab