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Limits – Class WorkDetermine the following limits using substitution.limx→5(x-5)92. limx→3311-x3. limx→-1x+2x2-1limx→07xx-85. limx→2ex26. limx→4(x-5)1009Determine the following limits using the graph of each function.limx→-1x+1x2-18. limx→1x+1x2-19. limx→0cos3x-13x limx→0sin5xsin8x11. limx→-1xx49720507747034861507747020002509652047561539370Use the graph to evaluate the following limits and functions. 12. 13. 14. 15. a. limx→1-fxa. limx→1-fxa. limx→1-fxa. limx→-2-fxb. limx→1+fxb. limx→1+fx b. limx→1+fxb. limx→-2+fxc. limx→1fx c. limx→1fx c. limx→1fx c. limx→-2fxd. f(1)d. f(1) d. f(1)d. f(-2)Use the piecewise functions to find the given limit.gx=4-2x;x<31-3x;x≥3 ; find limx→3g(x)17. gx=2x-4;x<34x-8x+1;x>3 ; find limx→3g(x)Limits – HomeworkDetermine the following limits using substitution.limx→6(x-5)9019. limx→3318+x220. limx→4x+2x2-1limx→17xx-822. limx→2ln(x2-3)23. limx→-4(x+4)1009Determine the following limits using the graph of each function.limx→-2x+2x2-425. limx→2x+2x2-126. limx→0cos2x-14xlimx→0sin8xsin2x28. limx→0+xx47625001333533337501333519335751333538100013335Use the graph to evaluate the following limits and functions. 29.30.31. 32. a. limx→5-fxa. limx→2-fx a. limx→4-fx a. limx→1-fxb. limx→5+fxb. limx→2+fx b. limx→4+fxb. limx→1+fxc. limx→5fx c. limx→2fx c. limx→4fx c. limx→1fxd. f(5)d. f(2)d. f(4)d. f(1)Use the piecewise functions to find the given limit.gx=4-2x;x<22x-8x;x≥2 ; find limx→2g(x)34. gx=3x-7;x<510x-2x+1;x>5 ; find limx→5g(x)247650053043Continuity – Class WorkState why the given points are points of discontinuity for f(x) and the type of discontinuity.f(-1)f(3)f(7)Identify any points of discontinuityfx=2-xx2-9x+1839. gx=e12x40. hx=2x-9lnxpx=x2+142. bx=secxFind any points of discontinuity. State whether the discontinuity is removable or not.y=3-x ; x≤2x-2x2-3x+2 ;2<x≤4xx+2 ; x>444. fx=x3-4x2-9x+36x3-5x2-2x+24;x≠412 ;x=4Find the value of k that makes f(x) continuous: fx=3k-2x;x<62kx+2 ;x≥6Find the value of k that makes f(x) continuous: fx=2k-2x;x<1k2-kx;1≤x<4k-x-2 ;x≥4Find the value of k and m that make f(x) continuous: fx=2m+x ;x≤-3mx+2k ; -3<x≤5kx+m-6 ;x>52352675-9525Continuity – HomeworkState why the given points are points of discontinuity for f(x) and the type of discontinuity.f(-1)f(3)f(7)Identify any points of discontinuityfx=2-xx2-4x+352. gx=e-3/x53. hx=2x-9ln(-x-2)px=x2-155. bx=cscxFind any points of discontinuity. State whether the discontinuity is removable or not.y=4-2x ; x≤2x2-4x+4x2-3x+2 ;2<x≤5x-2x-1 ; x>557. fx=x2+xx3+3x2+2x;x≠012 ;x=0Find the value of k that makes f(x) continuous: fx=3k+x;x<45k2x+2 ;x≥4Find the values of a and b that makes f(x) continuous: fx=2a-2bx;x<23a-4bx;2≤x<8-2a-4b-x ;x≥8Find the value of k and m that make f(x) continuous: fx=2m+x ;x≤52mx+3k ; 5<x≤7-k ;x>7Difference Quotient – Class WorkFind the average rate of change of the function over each of the given intervals.h(x)= 3x+4 for(a) [-2,5](b) [3,8]gx=4x3-2 for(a) [-3,0](b) [1,9]fx=e-x for(a) [0,4](b) [2,10]bx=sinx for(a) -π2,π2(b) 0,2πFind the rate of change at the given x.y=3x2-2x at x = 466. y=2x2-5x-7 at x = 2y=12x+1 at x = 268. y=2x3-2x2+3x+3 at x = 2y=2x2-5x-7 at x = a70. y=12x+1 at x = by=2x3-x2+6 at x = cDifference Quotient – HomeworkFind the average rate of change of the function over each of the given intervals.h(x)= 5x - 3 for(a) [-2,5](b) [3,8]gx=-2x3+4 for(a) [-3,0](b) [1,9]fx=ex for(a) [0,4](b) [2,10]bx=cosx for(a) -π2,π2(b) 0,2πFind the rate of change at the given x.y=-4x2-3 at x = 477. y=3x2+4x+5 at x = 2y=1x-1 at x = 279. y=3x3-4x2+5x+6 at x = 7y=3x2+4x+5 at x = a81. y=1x-1 at x = by=3x3-4x2+5x+6at x = cDerivatives – Class WorkUse the difference quotient to find the derivative of the given function at the given point.fx=2x3-4x ;at x=384. gx=1x;at x= -3Match the graph of the function to the graph of the derivative.4285615-190551589261103A4213526151765389038120081B42081450389038404C4210358615954787901270D26670025400Graph the derivative of the given function. Identify where the function is not differentiable.f(2)= 3 and f ‘(2)= 4, write the equation of the line tangent to f(x) at x=2. Write the equation of the normal line.If f(0)= 3 and f ‘(x)= 4, draw a graph of f(x), what is the equation for f(x)?fx=x2-2;x≤12x+4;x>1, draw the graph of f ‘(x).Derivatives – HomeworkUse the difference quotient to find the derivative of the given function at the given point.fx=4x3-2x2 ;at x=294. gx=12-x;at x= -4426037163852Match the graph of the function to the graph of the derivative.421513057785A389038961864313555-635B457751166579421513048260C 42545082550426496436195D74295066040Graph the derivative of the given function. Identify where the function is not differentiable.f(3)= 1 and f ‘(3)= -2, write the equation of the line tangent to f(x) at x=3. Write the equation of the normal line.If f(0)= 2 and f ‘(x)= 5, draw a graph of f(x), what is the equation for f(x)?fx=-2x2;x≤12x-2;x>1, draw the graph of f ‘(x). 369570076201Derivatives with a Calculator – Class Work43815051435Show that the following continuous functions are not differentiable.AB 369570051625560960024130AB Find the following derivatives at a point by using dydx=fx+.001-f(x).001.y=x23 at x = 8106. y=4x-6 at x = 5107. y=4x2-x at x = 3 Use the calculators built in derivative function to find the following derivatives.y=x53 at x = 2109. y=3x+7 at x = -4110. y=-32x2 at x = 5 The following function is continuous and differentiable. Find a and b.fx=a+bx, x<2ax2-x;x≥21524001250953525171123825Derivatives with a Calculator – HomeworkShow that the following continuous functions are not differentiable.AB 2952755295903724275481965AB Find the following derivatives at a point by using dydx=fx+.001-f(x).001.y=3x43 at x =9115. y=2x-4 at x = 2116. y=3x-x2 at x = 1 Use the calculators built in derivative function to find the following derivatives.y=x13 at x = 4118. y=3-x at x = -2119. y=-13x2 at x = 6The following function is continuous and differentiable. Find a and b.fx=2a-bx, x<4ax2-b+x;x≥4Derivative Rules: The Power Rule – Class WorkFind dydx.y=x3122. y=4x5-2x3+x123. y=5x7-5x+6+1x2y=x-3125. y=6x4+7x23126. y=4x+3xFind f'3.fx=5x3-4x128. fx=x2+x129. fx=x-4If f3=6, f'3=-2, g3=4, g'3=5, find h'(3) where h(x) is defined as follows:hx=fx-g(x)131. hx=7gx-5f(x)132. hx=4fx+2g(x) Find the equation of the tangent line to fx=4x2-3 at x=2.Given y=6x5-3x2+1, find d2ydx2.Given fx=1x, find f''(3).Derivative Rules: The Power Rule – HomeworkFind dydx.y=x5137. y=3x4-6x3+2x138. y=4x7-3x+7+3x2y=x143140. y=x-2x3141. y=7x-3+x2+5Find f'4.fx=5x3143. fx=6x+1x144. fx=x5-7x2+10x+9If f2=4, f'2=-5, g2=3, g'2=7, find h'(2) where h(x) is defined as follows:hx=fx+g(x)146. hx=3fx-5g(x)147. hx=8gx+f(x)Find the equation of the normal line to fx=2x3-3x at x=1.Given y=4x4-5x2+7x, find d2ydx2.Given fx=4x2, find f''(3).Derivative Rules: The Product and Quotient Rule – Class WorkFind dydx.y=x+2x2-3152. y=x(x2+1)153. y=x(1-x4)y=x2+1x2-1155. y=x2x+x156. y=3x+4x2+x+1Find f'3.fx=x-1x+1158. fx=2x+53x159. fx=x4-4x2-5If f3=6, f'3=-2, g3=4, g'3=5, find h'(3) where h(x) is defined as follows:hx=fxg(x)161. hx=f(x)g(x)162. hx=xf(x)Find the equation of the tangent line to fx=x32(2x4-3x+x-12) at x=4.Given y=2x-5x+3, find d2ydx2.Given fx=(x4-4)(x2+x+1), find f''(3).Derivative Rules: The Product and Quotient Rule – HomeworkFind dydx.y=4x3x-1167. y=x3+2x2+3x-1x168. y=x4+2x3x+1y=ax+bcx2-d170. y=(x2+x)(x+9)171. y=x2(3+x-1)Find f'4.fx=x+3x+1173. fx=3x-64x174. fx=4x3-xxIf f2=4, f'2=-5, g2=3, g'2=7, find h'(2) where h(x) is defined as follows:hx=fx+gxf(x)176. hx=g(x)f(x)177. hx=xg(x)Find the equation of the normal line to fx=3x-2x at x=4.Given y=4x+1x-2, find d2ydx2.Given fx=(4x2+9x)(3x-7), find f''(3).Derivative Rules: The Chain Rule – Class WorkFind dydx.y=(x2+9)4182. y=(7x-9)5183. y=12x+5y=(x4-x+2)-32185. y=1-x2186. y=x4+195Find f'3.fx=1+x2+5x33188. fx=x3+4x-2189. fx=x4-x3-15If f3=6, f'3=-2, g3=3, g'3=5, find h'(3) where h(x) is defined as follows:hx=f(x)3191. hx=7g(x)2192. hx=f(gx)Find the equation of the tangent line to fx=2x-55 at x=3.Given y=3x-7-2, find d2ydx2.Given fx=2x-55, find f''(3).Derivative Rules: The Chain Rule – HomeworkFind dydx.y=x2+3x-24197. y=5x2-x73198. y=4x5-xy=2x3-43200. y=3x-16201. y=3x2+8xFind f'4.fx=3x4-x-3-2203. fx=x+x34204. fx=x3+3x232If f2=2, f'2=-5, g2=3, g'2=7, find h'(2) where h(x) is defined as follows:hx=g(x)-3206. 2f(x)5207. hx=g(fx)Find the equation of the normal line to fx=6x-53 at x=1.Given y=3-4x4, find d2ydx2.Given fx=6x-53, find f''(3).Derivative Rules: Trigonometric Derivatives – Class WorkFind dydx.y=x2cosx212. y=9secx+12cotx213. y=sin(x2+4x)y=cot7(x5)215. y=tan3x+tan(x3)216. y=1+sinx1-sinxFind f'π4.fx=2sinx+3cosx218. fx=cscx-cotx219. fx=cosx1+sinxFind the equation of the tangent line to fx=sin(cosx) at x=π2.Given y=sin(x2), find d2ydx2.Given fx=3sinx+4cosx, find f''(π4).Derivative Rules: Trigonometric Derivatives – HomeworkFind dydx.y=x3cosx224. y=secxx2225. y=cot(4x2+9)y=sin(cossinx)227. y=tan(x+cosx)228. y=sinx4+cosxFind f'π6.fx=cosxsin2x230. fx=4tan2x231. fx=secxsinxFind the equation of the normal line to fx=cos(sinx) at x=π.Given y=cos(2x3), find d2ydx2.Given fx=xcosx, find f''(π).Derivative Rules: Combination – Class WorkFind dydx.y=2x+145sinx-13236. y=4x+73x-13y=x+1tanx-1238. y=1+x2+553y=x+x-134x-5240. y=1+1+2sinxFind f'3.fx=3x2-124x-3242. fx=3x5-x24x-33 243. fx=2x3-4x2-5x3If f3=6, f'3=-2, g3=3, g'3=5, find h'(3) where h(x) is defined as follows:hx=f(x)g(x)3245. hx=fxg(x)4246. hx=f(g(x)2Find the equation of the tangent line to fx=2x2-x4x3-x2 at x=1.Given y=3x-14x+53, find d2ydx2.Given fx=5x+232x-97, find f''(3).Derivative Rules: Combination – HomeworkFind dydx.y=(x+x5)42tanx-13251. y=x2-6x3x+72y=x3-4x25x2+3x253. y=5+sin5(x4+1)3y=3x+cos2x-4x3255. y=2x3-5x-233x-x2-4Find f'4.fx=2x-7x2-3x4257. fx=x2+2x+1x2-7258. fx=3x-x222x+852If f2=2, f'2=-4, g2=6, g'2=10, find h'(2) where h(x) is defined as follows:hx=g(x)f(x)5260. hx=xf(x)3261. hx=g(fx)4Find the equation of the normal line to fx=x2-35x-73 at x=2.Given y=9x-4-435-2x52, find d2ydx2.Given fx=5x+12x-5, find f''(3).Velocity, Speed, and Other Rates of Change – Class WorkWrite the equation for the area of a square in terms of its sides.Find the derivative of the area dAdsHow fast is the area changing when s=4.If the sides were measured in ft, what would the units of dAds be?Write the equation for the volume of a sphere in terms of its radius.Find the derivative of the volume dVdrHow fast is the volume changing when r=5.If the radius were measured in inches, what would the units of dVdr be?Write an equation for the perimeter of a right isosceles triangle in terms of its legs(s).Find the derivative of the perimeter dPdsHow fast is the perimeter changing when s=5.If the legs were measured in cm, what would the units of dPds be?A circle is inscribed in a square; write an equation for the area of the circle in terms of the square’s sides.Find the derivative of the area dAdsHow fast is the area changing when s=5.If the sides were measured in meters, what would the units of dAds be?t(seconds)012345678position(ft)182124252118161514.5Points for a differentiable function are given. Estimate the velocity at t=2 and t=6.From the table above, when does the particle have the greatest speed?44657591627505A swimming pool is drained for maintenance. The water remaining in the tank in gallons after t minutes is Wt=10,000(60-t)2. What is the rate the water is running out after 20 minutes? What is the average rate of change for the first 20 minutes? How long will the pool take to drain completely?The following graph shows the velocity of a particle. When is the particle’s speed the greatest?What is the particles acceleration at t=3?When is the particle speeding up?When does the particle change directions? stop?44672681099136The following graph shows the position of a particle. When is the particle’s speed the greatest?What is the particles velocity at t=3?When does the particle change directions? stop?Graph the particle’s velocity.Graph the particle’s speed.Velocity, Speed, and Other Rates of Change – HomeworkWrite the equation for the area of an equilateral triangle in terms of its sides.Find the derivative of the area dAdsHow fast is the area changing when s=4.If the sides were measured in ft, what would the units of dAds be?Write the equation for the volume of a cube in terms of its sides.Find the derivative of the volume dVdsHow fast is the volume changing when s=5.If the sides were measured in inches, what would the units of dVds be?Write an equation for the surface area of a sphere in terms of the circumference of its great circle.Find the derivative of the Surface area dSdCHow fast is the surface area changing when C=5.If the circumference were measured in cm, what would the units of dSdC be?A circle is circumscribed about a square; write an equation for the area of the square in terms of the radius.Find the derivative of the area dAdrHow fast is the area changing when r=5.If the radius were measured in meters, what would the units of dAds be?t(seconds)012345678position(ft)-5-124530-3-8Points for a differentiable function are given. Estimate the velocity at t=2 and t=6.From the table above, when does the particle have the greatest speed?45539271584960A swimming pool is drained for maintenance. The water remaining in the tank in gallons after t minutes is Wt=20,000(20-t)2. What is the rate the water is running out after 20 minutes? What is the average rate of change for the first 20 minutes? How long will the pool take to drain completely?The following graph shows the velocity of a particle. When is the particle’s speed the greatest? What is the particles acceleration at t=3?When is the particle speeding up?45542201329885When does the particle change directions? stop?The following graph shows the position of a particle. When is the particle’s speed the greatest? What is the particles velocity at t=3?When does the particle change directions? stop?Graph the particle’s velocity.Graph the particle’s speed.4357927-400050Applications – Class WorkA student went bungee jumping. The graph at the right shows his velocity.What was his velocity at t=3 seconds?What was his velocity at t=4 seconds? Was he speeding up or slowing down at t=5 seconds?What was his average rate of change from t=2 to t=5?Name another interval that has the same average velocity as d.When did the bungee cord engage?A particle moves along the x-axis and its position at any time 0≤t≤10 is st=3t2-4t+5.What is the particle’s initial position?What is the particle’s velocity at t=3 seconds?When does the particle switch directions?What is greatest distance the particle is from the origin?At t=3 seconds, is particle speeding up or slowing down?What is the total distance traveled by the particle?An above ground pool springs a leak. Water drains from the pool such that the height of the water in the pool at any given time, in minutes, is ht=4(360-x360)ft.How deep is the pool?How long till the pool is empty?What is the rate of change in the height at any given t?What is the average rate of change in height from t=30 minutes till t=60 minutes?A particle moves along the line y=2. Its velocity at any time 1≤t≤10 seconds is vt=-t+8tWhat is the particle’s velocity at t=2 seconds?When does the particle switch directions?At t=2 seconds is particle speeding up or slowing down?What is the fastest the particle is moving?The cost for making t tv’s is Ct=500+20t-0.2t2.What is the marginal cost, C'(t), of making 200 tv’s?What is the average cost of making 200 tv’s?Draw the graph of the marginal cost. When is marginal cost 0? 42867303362Applications – HomeworkA student went wake boarding. The graph at the right shows his velocity.What was his velocity at t=3 seconds? What was his velocity at t=4 seconds?Was he speeding up or slowing down at t=6 seconds?What was his average rate of change from t=5 to t=15?Was speed faster at t=2 or t=17?When did he let go of the rope?A particle moves along the line y=2 and its position at time 0≤t≤10 is st=-2t2+8t-4 What is the particle’s initial position?What is the particle’s velocity at t=3 seconds?When does the particle switch directions?What is greatest distance the particle is from the origin?At t=3 seconds is particle speeding up or slowing down?What is the total distance traveled by the particle?An above ground pool springs a leak. Water drains from the pool such that the height of the water in the pool at any given time, in minutes, is ht=3(180-x180)ft.How deep is the pool?How long till the pool is empty?What is the rate of change in the height at any given t?What is the average rate of change in height from t=30 minutes till t=60 minutes?A particle moves along the line y=0. Its velocity at time 1≤t≤10 seconds is vt=-2t+12t2What is the particle’s velocity at t=3 seconds?When does the particle switch directions?At t=3 seconds is particle speeding up or slowing down?What is the fastest the particle is moving?The cost for making t toasters is Ct=300+60t-0.01x2.What is the marginal cost, C'(t), of making 10,000 toasters?What is the average cost of making 10,000 toasters?Draw the graph of the marginal cost. When is marginal cost 0? Multiple Choicelimx→2x2-16x2-6x+8=0b. 1c. -12d. does not existe. ∞limx→4x2-16x2-6x+8=0b. 1c. 4d. does not existe. ∞limh→035-h2-4(5-h)-352-4(5)h=0b. -19c. -26d. -44e. does not existIf gx=x2+4x-21x-3;x≠39 ;x=3 , then which of the following statements is truei. g(3) existsii. limx→3g(x) exists iii. g(x) is continuousa. i is trueb. ii is true c. both i and ii are true d. all are true e. none are trueThe velocity of an object is vt=3t-100t2. What is the object’s average velocity from t=2 till t=5?30b. -30c. 10d. -10e. 7.3If gx=x2+(4+k)x+4kx+4;x<-45 ;x≥-4, find k so that g(x) is continuous-4b. 5c. 1d. 9e. does not existThe derivative of y=2tanx+5(4-sinx)2 isy'=8sec2x-2sinxsec2x-4sinx-10cosxy'=16secx-4sinxsecxy'=4-sinx8sec2x-2sinxsec2x-4sinx-10cosxy'=-16secx+4sinxsecxy'=4+sinx8sec2x+2sinxsec2x+4sinx+10cosxf4=6, f'4=8, g4=4, and g'4=-5, find h'(4) if hx=xf(gx)-160-154-403711575151130-3430The graph at right shows h'(x), h(x) is speeding up at23456Extended ResponseGiven the function hx=4x3+10x2-6x64x3+27,Find any points of discontinuity and describe the type of discontinuity that they are.If gx=x+6x, what type of discontinuity would hxg(x) have at x=0?If hx-1=f(x), what would be the discontinuities of f(x)?Given the function hx=4x2+10x2-6x64x3+27 and gx=x3+4x2+8xFind any horizontal asymptotes of h(x).Describe the end behaviors of g(x).If fx=hxg(x), find any horizontal asymptotes of f(x)hx=2x2+xm if x≤3xm2 if x>3Find the values of m that make h(x) continuous.Find the rate of change from h(1) to h(5).Describe the meaning of limx→3-hx=limx→3+hxThe table shows the position of a particle at given times.T012345s(t)68910913What is the particle’s approximate velocity at t=1?What is the particle’s average rate of change for t=0 to t=5?During what interval(s) is the particle’s velocity negative?Given the equation y=x-22x+4Show that dydx=x2+8x-20x2+8x+16Find the slope of the tangent line at the x-intercept (2,0).Find the slope of the normal line at the y-intercept (0,1)Find a point on the graph where the normal line is undefinedConsider the velocity graph.What is f2? What does it mean? What is f'(2)? What does it mean?2966967310696Is the object speeding up, slowing down, or neither at t=2? JustifyName any points of discontinuity. ................
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