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DERIVATIVES UNIT PROBLEM SETSPROBLEM SET #1 – Rate of Change ***Calculators Not Allowed***Find the average rate of change for each function between the given values:y=x2+4x+3 from x=-2 to x=3 fx=x3-x2+1 from x =-2 to x=2gx=sinx from x =π2 to x=πhx=2sinxcosx from x =π4 to x=3π4 rt=t2-9 from t=-5 and t=3 y=log10t from t=10 to t=100 y=2log5t from t =1 to t=25gt=et+5 from t =0 and t=1y=4-x2 from x =0 to x=3fx=x+2x-2 from x =3 to x=8Andrew is a physics student testing the rate of change of objects he can throw. Given his calculations, if he throws the baseball from the top of a hill, it follows the equation xt=-4.9t2+14.7t+25. He wants to know the average rate of change of the ball for each of the following time periods: (Calculator allowed)a) t=0 to t=1.5b) t=1.5 to t=3c) t=1 to t=3PROBLEM SET #2 – Slope of a Curve ***Calculators Not Allowed***For problems #1-8, find the limit of the function at the given point:a) Find the derivative of the function y=4x+1.b) What is the value of the derivative at x=1 ?a) Find f'(x) if the function is fx=2x2.b) What is the value at x=2 ?a) Find the derivative of gx=3x3+1.b) What is the value of g'(1) ?a) Find dydx of the function y=x+12b) What is the slope at x=-1 ?a) Find h'(x) if hx=2x3+3x2.b) What is the slope at x=-2 ?a) Find dfdr if fr=πr2.b) What is the slope at r=3 ?a) Find the derivative of y=x.b) What is the slope at x=2 ?Of the functions you have worked with, which type of functions have the same average rate of change as their instantaneous rate of change? (i.e. ΔyΔx=y') Try taking the average rate of changes of the examples above.a) Linearb) Quadraticc) Cubicd) Square RootPROBLEM SET #3 – Derivative Rules ***Calculators Not Allowed***Find the derivative of the function y=11.Find f'(x) if the function is fx=4x2.a) Find the derivative of gx=12x3-4x2+2x.b) What is the value of g'(-1) ?Find dydx for the function y=x+22.a) Find h'x if hx=2x3/4.b) What is the slope at x=16 ?a) Find dfdt if ft=3t .b) What is the slope at t=2 ?a) Find the derivative of y=2x+1(2x-1) .b) What is the slope at x=2 ?a) Find g'x if gx=x2+4x+12 .b) What is the slope at x=0 ?a) Find dydx if y=123x .b) What is the slope at x=8 ?PROBLEM SET #4 – Higher Order Derivatives ***Calculators Not Allowed***Find the 1st and 2nd derivative of the function y=20x .Find f'(x) and f''(x) of the function fx=13x2+4x .Find the 1st and 2nd derivative of gx=x+43 .Find dydx and d2ydx2 of the function y=x-42 .Find h'''(x) if hx=124x4-16x3Find d2fdt2 if ft=t .Find the derivative of y=3x .Find the 1st and 2nd derivatives of fx=xba) where b=2b) where b=3c) generically for all b>3Looking back at problem 8c above, we begin to see a pattern with derivatives. As we take derivatives, our exponents are used as coefficients and multiplied together. If we were to continue taking derivatives until the exponent is zero, we can see that the last coefficient is equal to the factorial (!) of the original power (any derivative after that will equal zero). If we stopped at any other time along that path, we would have a permutation of the exponents equal to aPd where “a” is the original exponent and “d” is the specific derivative. Using this method, find the following derivatives: (calculator OK)a) 8th derivative of x9b) 10th derivative of 2x10PROBLEM SET #5 – Trigonometry Rules ***Calculators Not Allowed***a) Find the derivative of the function y=sinx+cosx .b) What is the value of the derivative at x=π2 ?c) What is the value of the derivative at x=π6 ?a) Find f'(x) if the function is fx=1sinx+1cosx .b) What is the value of the derivative at π4 ?c) What is the value of the derivative at π3?a) Find the derivative of gx=3tanx-4cotx .b) What is the value of g'π6 ?a) Find dydx of the function y=1cotx+2cotx .b) What is the derivative at x=π4 ?a) Find h'(x) if hx=-cscx+secx .b) What is the slope at x=5π6 ?Find the first four derivatives of y=4sinx .Find the first four derivatives of gx=5cosx-sinxFind dmdx if mx=tan-1x+cot-1x .a) Find the derivative of y=2sin-1x .b) What is the value of the derivative at x=0.PROBLEM SET #6 – Product/Quotient Rule ***Calculators Not Allowed***Find f'(x) if the function is fx=x+2(x-2) .Find dydx of the function is y=x2+4x-3(3x2-10) .Find the derivative of gx=x(x4-3x2-10x+1) .Find h'(x) if hx=3x tanx .Find y' if y=cosx?sinx .Find f'(x) iffx=sin2x .Find the derivative of y=sinxcosx+cosxsinx (without using trig shortcuts).Find g'(x) if gx=x2+1x2-1 .Find dydt if y=et+1t3 .Using any prior rules, find the 1st and 2nd derivatives of y=tanx .**Show using product rule that the derivative of sin3x is 3sin2xcosx .PROBLEM SET #7 Derivatives Using Tables ***Calculators Allowed***xf(x)f’(x)g(x)g’(x)-2-1916-1119-1-610-24014-1-112-2-262-3-817Given hx=f(x)?g(x) find: h'(2)find: h'(1) Given jx=2x2?f(x) find: j'(0)find: j'(2)Given kx=f(x)g(x) find: k'(-1)find: k'(1)Given mx=gx2 find: m'(0)find: m'(-2)tg(t)g’(t)h(t)h’(t)0-106-1-91-31-352414-1-1/432-1.5-254081/21Given ft=g(t)?h(t)find: f'(1)find: f'(2)Given rt=g(t)h(t) find: r'(0)find: r'(3)Given st=ht2 find: s'(0)find: s'(4)Given gt=2t3h(t) find: g'(1)find: g'(2)PROBLEM SET #8 – Tangent/Normal Lines ***Calculators Not Allowed***For each question find the equations of the tangent & normal lines at the given value. fx=4x+7 at x=2fx=3x2-2x+1 at x=-3 y=3cosx+1 at x=π2gx=-4x-2 at x=4y=cosx?sinx at x=π4y=2x at x=-3hx=x2+1x2-1 at x=0fx=1x at x=2PROBLEM SET #9 – Derivatives of Log & e ***Calculators Not Allowed***Find the derivative of: fx=5lnx+1Find dydx for y=23x+15x+6xDifferentiate: y=3log4x+7log2xFind the derivative of: fx=x2+2xFind dydx for: y=5x+ex+lneFind y' for: y=lnx+x2log8x+x3Find the derivative of: fx=2ex+3x4ln2+2xFind dydx for: y=3xlog3x +3exDifferentiate: fx=5x+log7x+9lnx Find the derivative of : fx=3xlnx+3x2lnx+6PROBLEM SET #10 – Chain Rule ***Calculators Not Allowed***Find y' for : y=(5x2+3x)3 Given: fx=3x2x-173+15 Find f'(x)Given: fx=(4x3+2x2+ 12x)5 Find f'(x)Find y' for : y=3cos2x Differentiate: fx=(sin?(2x3-1))2 Given: fx=(3x4+x)(10x2-x)5 Find f'(x)Find dydx if : y=5x2-3x(3x7+2x6)4Differentiate: fx=(15x3-10x5)32 Given: fx=(7x+(3x-9)6)3 Find f'(x) Differentiate: fx=3tan?(8x)Find dydx if y=(-3x3+2x2+x)4 Find y' for : y=2x8(4x2-3x )2Given: fx=(7x2-x+15)-3 Find f'(x)Find dydx if y=-2sin4(3x-5) Differentiate: fx=(-4x2-6x)2(x5+5x)3 PROBLEM SET #11 – Derivs. of Inv. Functions ***Calculators Not Allowed***If f1=4 and f'1=6 , find f-1'(4) Find the derivative of the inverse of fx=x4 If f4=6 and f'4=5 find f-1'(6)If f2=7 and f'x=3x2+5x+12 find f-1 '(7)If f8=15 and f'x=x3+2x2+3x find f-1 '(15)Find the derivative of the inverse of fx=sinxIf f'x=2x+33x+2 and f3=6 find f-1 '(6)If f2=23 and f'x=5x+85x2+8 Find f-1 '(23) If fx=5x-7 Find f-1 '(8) If f1=5 and fx=2x4+2x3+2x2+2x+2 Find f-1 '(5)PROBLEM SET #12 – Contin. vs. Differentiability ***Calculators Not Allowed***If a function is differentiable on a given interval, it is also continuous.a. Trueb. FalseFor a function to be differentiable, it: (choose all that apply)a. Must have no discontinuitiesb. Can have discontinuitiesc. Must have no vertical tangent lines4029076153670d. Can have vertical tangent linese. Must not have cornersf. May have cuspsAt which values of x is f(x) not differentiable?4229100140335At which values of x is fx not differentiable?4124325131445At which values of x is f(x) not differentiable?PROBLEM SET #13 – Der. of Piecewise & Abs. Value ***Calcs. Not Allowed***a. What is the equivalent piecewise function for the following?b. What is the derivative?y=2x+3-4a. What is the equivalent piecewise function for the following?b. What is the derivative?f(x)=42x-2+5Find the derivative of the following function:y=3-x+2-1 x≤23x-2-1 x>2 Find the derivative of the following function:y=5x2-9x x<1lnx x≥1What values of k and m will make the function differentiable over the interval (-1,10)?fx=mx+4 -1≤x≤2kx2+3 2<x≤10What values of m and j will make the function differentiable over the interval (-3,9)?fx=mx2+5 -3≤x≤315+jx 3<x≤9What values of j and k will make the function differentiable over the interval (0,8)?fx=-j+2jx 0≤x≤12kx+15 1<x≤8For questions 8 – 10, choose all that apply:fx=x+23a. fx is continuous at x=-23b. f(x) is differentiable at x=-23c. f(x) is not continuous at x=-23d. f(x) is not differentiable at x=-23fx=4x2+x x≤12x4+x x>1a. fx is continuous at x=1b. f(x) is differentiable at x=1c. f(x) is not continuous at x=1d. f(x) is not differentiable at x=1fx=x2-4x+8 x≤32x-1 x>3a. fx is continuous at x=3b. f(x) is differentiable at x=3c. f(x) is not continuous at x=3d. f(x) is not differentiable at x=3PROBLEM SET #14 – Implicit Differentiation ***Calculators Not Allowed***Find dydx: y2+ x2=2x-5y3Find dydt: y=3t+2ty+t2Find dxdt: 15t=2x2+4xFind dydx: 3x2y+2x2+4y=5xFind dVdt: V=43πr3Find dydx: x4+xy4=4xFind dzdy: 3y3+3z2=5zFind dydx: sinx=2xcosy+2Find dθdt: 2sin2θ+cos2θ=2t Find dAdt: A=2πrh+2πr2 Find dydx: sin2x+cosy=5 Find the slope of the tangent line at the point (1,2) for: 3y2+5x3=17 Find the slope of the tangent line at the point (16,-1) for: 2y2+x-6=0 Find the slope of the tangent line at the point (1,1) for: 2x3=3x2-3xy+2 Find the equation of the tangent line at the point (4,8) for: 4x2-2y2=-16x Find the equation of the tangent line at the point (1,0) for: 1x+5=yx+6 Find the equation of the tangent line at the point (-3,3) for: xy2+y=8xLimits and Continuity- Answer KeysProblem Set #1 – Rate of Change 5 4 -2π -4π -12 190 16 e-1 13 -23 a. 7.35 b. -7.35 c. -4.9Problem Set #2 – Slope of a Curve a. y'=4 b. 4 a. f'x=4x b. f'2=8 a. g'x=9x2 b. g'1=9 a. dydx=2x+2 b. 0 a. h'x=6x2+6x b. h'2=12 a. dfdr=2πr b. 6π a. y'=12x b. 24 a. linearProblem Set #3 – Derivative Rules y'=0 f'x=8x a. g'x=36x2-8x+2 b. g'-1=46 dydx=2x+4 a. 324x b. 34 a. dfdt=32t b. 324 a. y'=8x b. 16 a. g'x=4x3+24x2+36x+8 b. g'0=8 a. dydx=163x2 b. 124Problem Set #4 – Higher Order Derivatives y'=20 y''=0 f'x=26x+4 f''x=26 g'x=3x2+24x+48 g''x=6x+24 dydx=2x-8 d2ydx2=2 h'''x=x-1 d2fdt2=-14t3 y'=133x2 a. f'x=2x f''x=2 b. f'x=3x2 f''x=6x c. f'x=bxx-1 f''x=b-1bxb-2 a. 362,880x b. 7,257,600Problem Set #5 – Trigonometry Rules a. y'=cosx-sinx b. y'π2=-1 y'π6=3-12 a. f'x=-cscxcotx+secxtanx b. f'π4=0 f'π3=-23+23 a. g'x=3sec2x+4csc2x b. g'π6=20 a. dydx=3sec2x b. 6 a. h'x=cscxcotx+secxtanx b. -23+23 y'=4cosx y''=-4sinx y'''=-4cosx y4=4sinx g'x=-5sinx-cosx g''x=-5cosx+sinx g'''x=5sinx+cosx g4x=5cosx-sinx dmdx=0 a. 21-x2b. 2Problem Set #6– Product/Quotient Rule f'x=2x dydx=12x3+36x2-38x-40 g'x=92x7/2-152x3/2-15x1/2+12x-1/2 h'x=13x-2/3tanx+x1/3sec2x y'=-sin2x+cos2x or y'=cos2x dfdx=2cosxsinx or y'=sin2x y'=sec2x+csc2x g'x=-4xx4-2x2+1 dydt=tet-3et-3t4 sec2x 2sec2xtanx must show workProblem Set #7 – Derivative Tables h'2=-29 h'1=16 j'0=0 j'2=-88 k'-1=1 k'1=-2 m'0=2 m'-2=-418 f'1=-18 f'2=-15 r'0=-96 r'3=-74 s'0=18 s'4=1 g'1=-8 g'2=-28Problem Set #8 – Tangent/Normal Linestangent: y-15=4(x-2) or y=4x+7normal: =y-15=-14x-2 or y=-14x+312 tangent: y-34=-20(x+3) or y=-20x-26 normal: y-34=120x+3 or y=120x+68320 tangent: y-1=-3x-π2 or y=-3x+3π+22 normal: y-1=13x-π2 or y=13x+6-π6 tangent: y+10=-(x-4) or y=-x-6 normal y+10=1(x-4) or y=x-14 tangent: y=12 normal: x=π4at x=-3:tangent: y=-2xnormal: y-6=12x+3 or y=12x+152at x=3:tangent: y=2xnormal: y-6=-12(x-3) or y=-12x+152 tangent: y=-1 normal: x=0 tangent: y-12=-14x-2 or y=-14x+1normal: y-12=4(x-2) or y=4x-152Problem Set #9 – Derivatives of Logs and e f'x=5x dydx=23xln23+15xln15+6xln6 y'=3xln4+7xln2 f'x=2x+2xln2 dydx=5xln5+ex y'=3x2+2xlog8x+xln8+1x f'x=2ex+24x3ln2+2xln2 dydx=3xln3(log3x+3ex)-3x1xln3+3ex(log3x+3ex)2 f'x=5xln5+1xln7+9x f'x=3xln3lnx+3xx+6xlnx+3xProblem Set #10 – Chain Rule3(5x2+3x)2(10x+3)3(2x-17)3+18x(2x-17)254x3+2x2+12x4(12x2+4x+12)-6cosx sinx12x2sin2x3-1cos?(2x3-1)(12x3+1)(10x2-x)5+5(3x4+x)(10x2-x)4(20x-1)(10x-3)(3x7+2x6)4-4(5x2-3x)(3x7+2x6)3(21x6+12x5)(3x7+2x6)832(15x3-10x5)12 (45x2-50x4) or 32(45x2-50x4)15x3-10x5 9(7x+(3x-9)6)2 (7+6(3x-9)5) 24sec28x 4(-3x3+2x2+x)3 -9x2+4x+1 16x7(4x2-3x)2+4x8(4x2-3x)(8x-3) -42x+3(7x2-x+15)4 -24sin33x-5 cos?(3x-5) 2(?4x2-6x)(-8x-6)(x5+5x)3+3(-4x2-6x)2(x5+5x)2(5x4+5)Problem Set #11 – Der. of Inverse Fns.16144x3 or 14x-3415134166411-x211914915120Problem Set #12 – Continuity vs. Diff.Truea, c, ex=-4, x=-1x=-1, x=4, x=6x=-4, x=-3, x=-2, x=3Problem Set #13 – Derivatives of Piecewise & Absolute Value Functionsy=-2x-10 x≤-32x+2 x>-3 y'=-2 x<-32 x>-3 y=-8x+13 x≤18x-3 x>1y'=-8 x<18 x>1y'=-3 x<23 x>2y'=10x-9 x≤11x x>1k=-14 , m=-1j=-203, m=-109j=-15, k=-15a, dc, d a, bProblem Set #14 – Implicit Differentiationdydx=2-2x2y+15y2dydt=3+2y+2t1-2tdxdt=154x+4dydx=10y2-8xy2-3y8y2-3xdVdt=4πr2drdtdydx=4x3+y4-4-4xy3dzdy=9y25-6zdydx=cosx-2cosy-2xsinydθdt=1sinθcosθ dAdt=2πhdrdt+2πrdhdt+4πrdrdt dydx=2cos2xsiny m=-54 m=132 m=-1 y-8=32x-4 or y=32x+2 y=-x+1 y-3=117x+3 or y=117x+5417 ................
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