CALCULUS First Principles - La Citadelle

1 CALCULUSF irst Principles1. Use the First Principles formula limh 0f(x+h) f(x)hto find the derivative function for the folowing func-tion:f(x) = x2+ 2x32. Use the First Principles formula limh 0f(x+h) f(x)hto find the derivative function for the folowingfunction:f(x) = 2 +x3. Use the First Principles formula limh 0f(x+h) f(x)hto find the derivative function for the folowingfunction:f(x) = 1 2x4. Use the First Principles formula limh 0f(x+h) f(x)hto find the derivative function for the folowingfunction:f(x) = 1 + 2x45. Use the First Principles formula limh 0f(x+h) f(x)hto find the derivative function for the folowingfunction:f(x) = x+ 3x46.

2 Use the First Principles formula limh 0f(x+h) f(x)hto find the derivative function for the folowingfunction:f(x) = x+ 3x27. Use the First Principles formula limh 0f(x+h) f(x)hto find the derivative function for the folowingfunction:f(x) = 3x x48. Use the First Principles formula limh 0f(x+h) f(x)hto find the derivative function for the folowingfunction:f(x) = 2x+ 2x49. Use the First Principles formula limh 0f(x+h) f(x)hto find the derivative function for the folowingfunction:f(x) =x3+ 2x410. Use the First Principles formula limh 0f(x+h) f(x)hto find the derivative function for the folowingfunction:f(x) = 2x+ 3x2 Answers:1. 2x+ 6x22.

3 13. 24. 8x35. 1 + 12x36. 1 + 6x7. 3 4x38. 2 + 8x39. 3x2+ 8x310. 2 + 6xc 2009 La Citadelle1 of PrinciplesSolutions: First , we need to know the following algebraic identities:a2 b2= (a b)(a+b)a3 b3= (a b)(a2+ab+b2)a3+b3= (a+b)(a2 ab+b2)a4 b4= (a b)(a+b)(a2+b2)Second, we need to understand the computation of the following fundamental limits:L0= limh 01 1h= limh 00h= limh 00 = 0L1= limh 0(x+h) xh= limh 0hh= limh 01 = 1L2= limh 0(x+h)2 x2h= limh 0[(x+h) x][(x+h) +x]h= limh 0h(2x+h)h= limh 02x+h= 2xL3= limh 0(x+h)3 x3h= limh 0[(x+h) x][(x+h)2+ (x+h)x+x2]h= limh 0h[(x+h)2+ (x+h)x+x2]h= limh 0(x+h)2+ (x+h)x+x2= 3x2L4= limh 0(x+h)4 x4h= limh 0[(x+h) x][(x+h) +x][(x+h)2+x2]h= limh 0h[(x+h) +x][(x+h)2+x2]h= limh 0[(x+h) +x][(x+h)2+x2] = (2x)(2x2) = (x) = limh 0f(x+h) f(x)hIUse substitution:= limh 0[ (x+h)2+ 2(x+h)3] [ x2+ 2x3]hIGroup the like terms and factor:= limh 0(x+h)2 x2h+ 2 limh 0(x+h)3 x3hIIdentify the fundamental limits:= L2+ 2L3 ISubstitute the values of the fundamental limits:= (2x) + 2(3x2)ISimplify.

4 = 2x+ (x) = limh 0f(x+h) f(x)hIUse substitution:= limh 0[ 2 + (x+h)] [ 2 +x]hIGroup the like terms and factor:= 2 limh 01 1h+ limh 0(x+h) xhIIdentify the fundamental limits:= 2L0+L1 ISubstitute the values of the fundamental limits:= 2(0) + (1)ISimplify:= 1c 2009 La Citadelle2 of (x) = limh 0f(x+h) f(x)hIUse substitution:= limh 0[1 2(x+h)] [1 2x]hIGroup the like terms and factor:= limh 01 1h 2 limh 0(x+h) xhIIdentify the fundamental limits:=L0 2L1 ISubstitute the values of the fundamental limits:= (0) 2(1)ISimplify:= (x) = limh 0f(x+h) f(x)hIUse substitution:= limh 0[ 1 + 2(x+h)4] [ 1 + 2x4]hIGroup the like terms and factor:= limh 01 1h+ 2 limh 0(x+h)4 x4hIIdentify the fundamental limits:= L0+ 2L4 ISubstitute the values of the fundamental limits:= (0) + 2(4x3)ISimplify:= (x) = limh 0f(x+h) f(x)hIUse substitution:= limh 0[ (x+h) + 3(x+h)4] [ x+ 3x4]hIGroup the like terms and factor:= limh 0(x+h) xh+ 3 limh 0(x+h)4 x4hIIdentify the fundamental limits:= L1+ 3L4 ISubstitute the values of the fundamental limits:= (1) + 3(4x3)ISimplify:= 1 + (x) = limh 0f(x+h) f(x)hIUse substitution:= limh 0[ (x+h) + 3(x+h)2] [ x+ 3x2]hIGroup the like terms and factor:= limh 0(x+h) xh+ 3 limh 0(x+h)2 x2hIIdentify the fundamental limits:= L1+ 3L2 ISubstitute the values of the fundamental limits:= (1) + 3(2x)ISimplify:= 1 + (x) = limh 0f(x+h) f(x)hIUse substitution.

5 = limh 0[3(x+h) (x+h)4] [3x x4]hIGroup the like terms and factor:c 2009 La Citadelle3 of Principles = 3 limh 0(x+h) xh limh 0(x+h)4 x4hIIdentify the fundamental limits:= 3L1 L4 ISubstitute the values of the fundamental limits:= 3(1) (4x3)ISimplify:= 3 (x) = limh 0f(x+h) f(x)hIUse substitution:= limh 0[ 2(x+h) + 2(x+h)4] [ 2x+ 2x4]hIGroup the like terms and factor:= 2 limh 0(x+h) xh+ 2 limh 0(x+h)4 x4hIIdentify the fundamental limits:= 2L1+ 2L4 ISubstitute the values of the fundamental limits:= 2(1) + 2(4x3)ISimplify:= 2 + (x) = limh 0f(x+h) f(x)hIUse substitution:= limh 0[(x+h)3+ 2(x+h)4] [x3+ 2x4]hIGroup the like terms and factor:= limh 0(x+h)3 x3h+ 2 limh 0(x+h)4 x4hIIdentify the fundamental limits:=L3+ 2L4 ISubstitute the values of the fundamental limits:= (3x2) + 2(4x3)ISimplify:= 3x2+ (x) = limh 0f(x+h) f(x)hIUse substitution:= limh 0[2(x+h) + 3(x+h)2] [2x+ 3x2]hIGroup the like terms and factor:= 2 limh 0(x+h) xh+ 3 limh 0(x+h)2 x2hIIdentify the fundamental limits:= 2L1+ 3L2 ISubstitute the values of the fundamental limits:= 2(1) + 3(2x)ISimplify:= 2 + 6xc 2009 La Citadelle4 of