Finl Exm Study Guide Includes. Integrls & Antiderivtive Rules 2. Definite Integrls (Integrls with bounds) 3. Are Between Two Curves - Region Bounded by Two Curves 4. Consumer nd Producer Surplus. U-Substitution. Integrls & Antiderivtives A. s Remember to lwys stop nd look for rewrites before you strt trying to tke the ntiderivtive. These re the three rewrites you need to look for.. Rdicls root root x p r x p r 3 x x 3 Squre root x b. x on the bottom x 2 number x p x 8 number x power x p x 8 c. Both number root r x p 3 x st Rdicl number root x p r x 3 2nd x on the bottom number x power root x p r x 3
B. Power Rule This is the min rule tht you will use when finding n ntiderivtive. Rule: : Add to the power. : Divide by the new power. Finl 2 Add to the power. Divide by the new power. First pply s Second pply Power Rule x x + x + + 3 x x 6 x 6 6 x 6 6 +C : Rdicl x 3 : x on bottom x 3 : Power Rule Add to the power. x 3 + x 2 3 Step 4: Divide by the new power. Divide by the new power. x 3 + x 2 3 3 + 2 3 Finl x 2 3 2 3 + C
C. Specil Cses. Nturl Log = ln(x) number x x x number ln(x)+ C ln(x)+ C ln(x) + C b. e to power e ( number x) number ie( number x) + C e ( x) ie( x) + C e ( x) ie( x) + C 2. Definite Integrls (Integrls with bounds) 3 3 x + x 3 + 7x + Hit Y= on your clcultor nd input the eqution into Y The cube root button is under the Mth button. Y= 3 x +x 3 +7x+ Mke sure your x-window is lrge enough to tke in your bounds. 3 & in this cse. Hit: 2nd >CALC >#7 f (x) -It will then sk you to enter lower bound. Type 3 nd hit Enter. -It will then sk you to enter n upper bound. Type nd hit Enter. UpperBound LowerBound Clcultor will output your nswer t the bottom of the screen. f (x) = 749.673
3. Are Between Two Curves - Region Bounded by Two Curves Eqution You will need to setup this eqution to find the nswer. b Are = Top (Bottom) Step 4 Step Step 6 Hit Y= on your clcultor nd input the equtions into Y & Y2 Find the upper nd lower bounds for your sitution. Adjust your window to be ble to see the region bounded by the two curves. Hit: 2nd >CALC ># Intersect You will need to do this process twice to find the two intersects. Bring together your results from nd Step 3 to crete your eqution. Enter your new eqution into Y Find the re of the region bounded by the grphs of the functions y = x nd y = x 4 in the first qudrnt (where x nd y ). Hit: 2nd >CALC >#7 f (x) The nswer will be t the bottom of the screen. Y= x Y2= x 4 Identify which grph is your Top eqution (bove), nd which one is your Bottom eqution (below). -It will sk you to select the First Curve. Just hit Enter. -It will sk you to select the Second Curve. Just hit Enter. -It will sk you for guess. Scroll closer to one of the interests nd hit Enter x (x 4 ) Y= x x 4 -It will sk you for Lower Bound, enter. -It will sk you for n Upper Bound, enter. f (x)=.3
4. Consumer nd Producer Surplus Equtions You will need to setup these equtions to find the nswers. Consumer Surplus Producer Surplus D(x) ( i P) (i P) S(x) Equilibrium Point (,P) S(x) = D(x) Find the Consumer nd Producer Surplus given D(x) = 8 x 2 S(x) = x 2 + 4x + Hit Y= on your clcultor nd input the supply nd demnd equtions into Y & Y2 Adjust your window to be ble to see the intersections of the two grphs. Y=8 x 2 Y2 = x 2 + 4x + Note: You will see two intersections one in the positive x nd one in the negtive x. Alwys choose the positive. Find the intersection of the two grphs. Hit: 2nd >CALC ># Intersect -It will sk you to select the First Curve. Just hit Enter. -It will sk you to select the Second Curve. Just hit Enter. -It will sk you for guess. Scroll closer to one of the interests nd hit Enter You should find the intersect (,6)=(,P) this is your equilibrium point. Step 4 Setup your Consumer Surplus Eqution D(x) ( i P) 8 x 2 (i6) Step Plug the Demnd Eqution into Y. Y = 8 x 2 Hit: 2nd >CALC >#7 f (x) Lower bound enter: Upper bound enter: = 363.33 Tke from Step nd subtrct off (i6) = (28) Step 6 363.33-28 = 83.33 Consumer Surplus = 83.33
Step 7 Setup your Producer Surplus Eqution (i P) S(x) (i6) x 2 + 4x + Step 8 Plug the Supply Eqution into Y. Y= x 2 + 4x + Hit: 2nd >CALC >#7 f (x) Lower bound enter: Upper bound enter: = 46.67 Tke from Step 8 nd subtrct it from (i6) = (28) Step 9 28-46.67 = 33.33 Producer Surplus = 33.33
. U-Substitution (2x2 3x + 7) 3 (4x 3) Step 4 Step Step 6 Step 7 Select your u. Priority List for selecting u ) Inside Piece (like in chin rule problem) 2) ln(x) 3) Bottom of Frction 4) Highest Power of x Tke the derivtive of your u. Mke sure to use this specific derivtive nottion. Solve for. (i.e. get by itself) Multiply both sides by Plug the u from & the from bck into originl eqution. Simplify the Eqution Cncel out the (4x 3). All your x s should cncel out every time so tht ll you re left with is n eqution with u s. Apply the Power Rule Plug the u from bck in. Divide both sides by (4x 3) u = 2x 2 3x +7 du = 4x 3 i du = (4x 3)i du = (4x 3)i du (4x 3) i = (4x 3) (4x 3) du (4x 3) = (u) 3 du (4x 3) (4x 3) (u)3 du (u) 4 4 + C (2x 2 3x + 7) 4 + C 4