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# Calculus MAT 021B

UCD

GPA 3.88

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This 23 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 021B at University of California - Davis taught by Qinglan Xia in Fall. Since its upload, it has received 33 views. For similar materials see /class/187344/mat-021b-university-of-california-davis in Mathematics (M) at University of California - Davis.

## Reviews for Calculus

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Date Created: 09/08/15

Review for Midterm 1 Information for Midterm l 0 Where Here in this classroom a When This Friday October 172008 on lecture time 1210 1100pm a Who You of course a What Contents of Section 48 and Chapter 5 This is a closed booknotesfriends exam Totally 4 pages 6 problems No calculator is allowed A previous exam for the last year has been posted on the course webpage You are encouraged to take it rst and then compare your answers with the provided solutions Helpful resources a This Review a Previous Exam posted on the course webpage o Homework In particular PAPER HOMEWORK o Lecture Notes a Textbook 0 Online learning materials Some basic conceptsmethodstheorems o Antiderivatives Inde nite integral 0 Initial value problem 39 Partition of an interval and the norm of a partition 0 Riemann Sums In particular upper sum and lower sum 0 Sigma notation 0 Average of an integrable function a de nite integral a The substitution rule for both inde nite integral and de nite integral a Fundamental theorem of Calculus o Area between two curves o Calculate 525 hlttgtdt 1 Find antiderivatives of f Le nd f f Example 1 2x2 3x38d T56 1 Find antiderivatives of f Le nd f f Example 1 2x2 3x38d T56 2x2 3x38 dx 2x32 3x18dx 45 247 57x 2 Evaluate inde nite integrals using the substitution rule Example 2 2 xe3x xsinlt3x2d 2 Evaluate inde nite integrals using the substitution rule Example 2 2 xe3x xsinlt3x2d Let u 3x2 then du 6xdx Thus 2 LE 3 xsinlt3x2dx u y du e sumo 6k cosu C l 63552 cos3x2 C Example 3 1 d Example 3 100 d5 m d Letu 1 then du 2 Thus 100 d5 W 2 du U100 2 99 C ggu 2 E1 rs C 3 Evaluate de nite integrals using the substitution rule Example 4 l6 dx 2 3xxlrm 3 Evaluate de nite integrals using the substitution rule Example 4 l6 dx 2 3xxln Let u lnx then du d m Also When m 2u ln2 and when x 16 u ln 16 Thus 16 din lnl6 du 9mg am M gm m 4 Solve the differential equation f with the initial condition yltx0 3J0 Example 5 Find the curve in the xy plahe that passes through the point 6 LC e4139 0 and whose slope at each point is 4 Solve the differential equation f with the initial condition yltx0 3J0 Example 5 Find the curve in the xy plahe that passes through the point 6 LC e4l39 0 and whose slope at each point is We need to solve the differential equation Solve the differential equation f with the initial condition yltx0 3J0 Example 5 Find the curve in the xy plahe that passes through the point 6 LC e4l39 0 and whose slope at each point is We need to solve the differential equation with the initial condition ylt0gt So Let u e25 Then du 2e2dm Therefore d l l y E arotanu C E arotanlte2x C Substitute the initial condition 310 we have 1 g iarotanlt1gt C g 0 Thus C Therefore the curve is y arotane2x g 5 Calculate area between curves Example 6 Setup the area between the xaxts and the graph of the curve y x2 x 6 over the interval 10 10 in terms 0fthtegrals 5 Calculate area between curves Example 6 Setup the area between the xaxz s and the graph of the curve y x2 x 6 over the interval 10 10 in terms 0fthtegrals 10 A x2 x 6dx 10 As 2 x 6 x 2x 3 the area can also be expressed as 2 3 10 x2 x 6dx x2 x 6d xZ w 6dx lO 2 3 The exact area is 17 but you are not required to nd this value The area is 2 10 Example 7 Find the area between curves x smltygt ey tan5y and 2 10 x ey tan5y over the interval 0 g y 3 7T 2 10 Example 7 Find the area between curves x sinltygt ey tan5y and x e92tan5910 over the interval 0 g y 3 7T The area is A 7TltSmltygt eyQ tan5y10gt eyQ tan5y10gtdy Therefore 0 7T A sinydy 008yg11 2 0 6 Average value of an integrable function 1 5 Average b aa fltxgtdx Example 8 Find the average of the function y sinlt3xgt over 0 7T 6 Average value of an integrable function 1 b b aa fltxgtdx Example 8 Find the average of the function y sinlt3xgt over 0 7T Average The average value is 1flew 1 I ll sm x 7T 0 7T 3 0 7T3 3 37T

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