35. Formula Proof Problem I

lf f(x) = k.g(x), then f'(x) = k.g'(x).Proof:

r,(*) =;Tolllr*-lc)=litjij:@h-ro h

=limk.@h+o h

= k.g'(x), Q.E.D.

36. Formula Proof Problem lllf f(x) = x5, then f'(c) = 56+.Proof:

f'(c)= rim !0*nCI.. x"-c"

=ltm-x--rc X - c,,- (x : cXl4 + x3c .+ r3ql t-rs:l-g1)

=llm-

= jd o, + x3c + ,r", *f,i *""0;

=&+c4+c4+c4+c4= 5c4, Q.E.D.

37. Derivative of a Power Formulalf f(x) = xn, then f'(x) = ryn-1.Proof:

lr-l=,1gf0aiFa

38. Derivative of a Sum of n Functions Problemllyn - u1 + u2 + u3+. . . + un, wherethe u; aredifferentiable functions of x, prove thatYn' = u1' + u2' + u3' +. . . + un' for a//integers n 22.Proof:Anchor: For n = 2, yz= u1 + u2.:. Y2' = u1' + u2' by the derivative of a sum of fwofunctions property, thus anchoring the induction.lnduction Hvoothesis:Suppose that for n =k> 2,

Yk'= ut'+ u2'+ u3'+. . . + u1'.Verification for n = k + 1:

Letyl*1 =ul *U2+ u3+'.. + Uk+ uk+1.Thenyp*1 -(ur + u2+ u3+.. . + u1) + u1*1, which isasum of tulo terms..'. Yk+r'- (ur + u2 + u3 +. . . + up)'+ up*1'

by the anchor= ul '+ u2'+ u3'+.. . + up' + u1*1',

which completes the induction..'. Yn'= u1'+ u2'r u3't . ' .* Un' for a//integers n ) 2'Q.E.D.

39. lntroduction to Antiderivativesa. f'(x) = 3x2- 10x + 5 + f(x) = xs- 5x2 + 5xb. g(x) = l(x) + 13 is also an answer to part a, because it

has the same derivative as f(x). The derivative of aconstant is zero.

c. The name antiderivative is chosen because it is aninverse operation of taking the derivative.

3. x = -t3 + 13t2 - 351 + 27. Graph.The object starts out alx=27 ft when t = 0 sec. ltmoves to the left to x = 0.16 ft when t = 1.7 sec. lt turnsthere and goes to the right to x = 70 ft when t = 7 sec. ltturns there and speeds up, going to the left for allhigher values of t.

-k. limh+0

g(x+h)-s(x)h

xn + m(rF-1h + hrrtlrgf + +tfl-f= lim

h-+0

= lim (nxn-1 + ]n(n-1)xn-2h +. .. + hn-1 )h-+0

=p1rFl+0+0+...+0= nXr1, which is from the second term in the

binomial expansion of (x + h)n, Q.E.D.

Problem Set 3-5. poges 102 to 104 Displocement. Velocity. ond Accelerotion

45. (d/dx)(3x + 5) = 3 Q6. f(3) = 45

Q1. No values of tQ3. y'= -51x-a

Q7. f'(3) = 30Q9. Epsilon

1.y=sta-3F/+71u=ff= 2gf -7.211.a *

2.y=0.3t4-5tu=ff= -1.2trs-5,a=

Q2. dy/dx = 10xaa. f'(x) = 1.7x0.7

Q8. Limit = 45Q10. Definite integral

z,a=S= 60t2-10.08t0'4

#= u.u

Turnsatt=7,x=76.

Problem Set 3-5 Solutions Monuol 33