DOC PREVIEW
UW-Madison MATH 221 - Math 221 Review Sheet

This preview shows page 1 out of 4 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 4 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 4 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Math 221 Review SheetBNFall 2000Disclaimer: This may not be comprehensive! If a topic is not onthis review sheet, it might still be on the exam!Advice: Learn how to do the problems you missed on the earlierexams and quizzes. Study the examples in the text. Many haveappeared on earlier exams.1. Write the formula which defines the derivative f0(a) of the function f(x) atthe point x = a.2. (i) Define the terms differentiable function and continuous function. (ii) Givean example of a continuous function which is not differentiable. (iii) Prove thata differentiable function is continuous. (Use the definitions you gave in part (i).You may use without proof the theorem that the limit of a product is the productof the limits.)3. State the Mean Value Theorem.4. State the Intermediate Value Theorem.5. (i) Complete the definition: The Taylor polynomial of the function f ofdegree n centered at a isP (x) =nXk=0. . .(ii) In what sense is the Taylor polynomial close to the function f?6. Let f(x) be a function defined on the interval a ≤ x ≤ b. A Riemann sumfor f on the interval [a, b] is an expression of formS =nXk=1f(ck)(xk− xk−1)where a = x0< x1< ··· < xn= b and xk−1≤ ck≤ xkfor k = 1, 2, . . . , n.Complete the sentence: When the quantity is small, the Riemannsum is close to .17. (i) Find the Taylor Polynomial of degree 5 for f(x) = exat a = 0. (ii) Esti-mate the error if you use part (i) to approximate e−1.8. Find the polynomial of degree 4 which best approximates the function f(x) =tan x near the point x =π4.9. Graph, indicate limits at x = ±∞ and the one sided limits where the functionis undefined, and label all maxima, minima, and points of inflection:(i) y = xe−x2. (ii) y =1(x − 1)(x − 2).10. Find the limits:(i) limx→∞exx5(ii) limx→∞x22x(iii) limx→0x ln x(iv) limx→a2x− 2ax − a(v) limx→0ex− 1sin x(vi) limx→1xa− 1xb− 1(vii) limx→01x− csc x (viii) limx→∞(ln x)3x2(ix) limx→1ax− 1bx− 111. The altitude of a triangle is increasing at a rate of 1 cm/min while the areais increasing at a rate of 2 cm2/min. At what rate is the base of the trianglechanging when the altitude is 10 cm, and the area is 100 cm2.12. Find the maximum value and the minimum value of f(x) =ln xxon [1, 3].13. Find the point of the graph of x+y2= 0 that is closest to the point (−3, 0).14. Find the volume of the largest right circular cone that can be inscribed ina sphere of radius 5.15. Find the area under the curve y = 2x + 1, from a = 0 to b = 5, usingRiemann sums. Hint:nXk=1k =n(n + 1)2.16. EvaluateZ11x2cos x dx.17. Given thatZ94√x dx =383, what isZ49√t dt?18. Use a Riemann sum with four terms to find a number slightly larger thanZ10p1 + x3dx. Illustrate this with a picture.19. Finddydxif y =Zπx2sin ttdt.220. Find the interval on which the curve y =Zx011 + t + t2dt is concave up-ward.21. Verify thatZx2sin x dx = −x2cos x + 2Zx cos x dx22. Evaluate:(i)Z2ax + b√ax2+ bx + cdx (ii)Zsin(3A) − sin(3x) dx(iii)Ztan−1x1 + x2dx (iv)Zsec2(3θ) dθ23. Find the values of c such that the area of the region bounded by y = x2−c2and y = c2− x2is 576.24. Set up integrals for the volume of the solid obtained by rotating the regionbounded by the given curves about the given line:(i) y = ln x, y = 1, x = 1, about the x-axis.(ii) 2x + 3y = 6, (y − 1)2= 4 − x, about x = −5.(iii) y = cos x, y = 0, x = 0, x =π2, about y = 1.(iv) y =11 + x2, y = 0, x = 0, x = 3, about y-axis.25. The temperature of a metal rod, 5 m long, is 4xoC at a distance x metersfrom one end of the rod. What is the average temperature of the rod?26. IfZx20f(t) dt = x sin(πx) find f(4).27. The displacement in meters of a particle moving in a straight line is givenby s = t2− 8t + 18, where t is measured in seconds.28. Simplify:(i) tan−1(√3) (ii) sin[sin−1(13) + sin−1(23)] (iii) sin(tan−1x)29. Find the equation of the tangent line to the curve y = tan−1(3x − 2) atx = 130. Find the derivative:(i) f (x) = cos−1(sin x)(ii) g(x) = x tan−1x(iii) h(x) = (sin−1x)(ln x)31. Find the length of the curve y = ln(cos x) from 0 ≤ x ≤π4.32. Find the area and centroid of the region bounded by the given curves:3(i) y = x2, y = 0, x = 2. (ii) y = x, y = 0, y =1x, x = 2.33. Given thatddy(ln y) =1y, prove thatddx(ex) = ex.34. Finddydxif tan−1x + sin−1y = ln(xy)35. If x[f (x)]3+ xf(x) = 6 and f(3) = 1, find f0(3).36. Find z ifdzdx= e−x2/2x and z = 1 when x = 0.37. A population Y is growing exponentially in time. (i) If Y = 100 at timet = 0 and Y = 300 at time t = 2 what is Y at time t = 5? (ii) Express dY /dtas a function of Y


View Full Document

UW-Madison MATH 221 - Math 221 Review Sheet

Download Math 221 Review Sheet
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Math 221 Review Sheet and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Math 221 Review Sheet 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?