l.

I

d. Line: y = 1.4x + 1.4. GraPh h. f (32) = $1szlars = 201.6 exactly'

Numerical derivative is equal to or very close to201.6.

i. Graph (Dotted line)(Function is y = g.5xz - x + 4')

". u =# or x'(t).

" =$or.v'(t), a =S orx"(t)

b. Soaceship Problemi. y = -{.01t3 + 0.9t2 - 251 + 25o

u=S=-0.03t2+1.81-25

" =S = -0.06t + 1.8

ii. a(15) = -0.06(15) + 1.8 = 0.9 (km/sec)/secv(15) = -{.03(152) + 1'8(15) - 25

= 4.75 km/secThe spaceship is slowing down at t = 15 becausethe vdlocity and the acceleration have oppositesions.

iii. v: -o.o3t2 + 1.8t - 25 = 0By souve or quadratic formula,t = 21 .835... or t = 38'164.'. .

The spaceship is stopped at about 21.8 and 38'2seconds.

iv. y = -0.01t3 + 0.9t2 - 25t + 250 = 0By TRAcE ot g91Yg, t = 50'v(50) = -10Sinc'e tne spaceship is moving at 10 km/sec when

it reaches the surface, it is a crash landing!

R6. a. GraPh.

e.l.

The line is tangent to the graph.Yes, f does haire local linearity at x = 3' Zoom.ing in

on the point (3, 5.6) shows that the graph looks

more and more like the line.

Yr = t' - 4x3 - 7x2 + 34x - 24. Graph-R3. a.

b. yz = numerical derivative. See graph in part a'

". i-n" yr graph has a high point or a low point at each

x-vaiue where the Y2 graPh is zero.

d. Leaky Tire Problemp(t)=35(0.9)t.GraPh

Take the numerical derivative at t = 3, 6, and 0.

p'(3) = -2.688...bicreasing at about 2.69 psi/hr when t = 3'p'(6) = -1.e59...bdireasing at about 1.96 psi/hr when t = 6'p'(0) = -3.687...bdireasing at about 3'69 psi/hr when t = 0'

The units are Psi/hr.The sign of the pressure change is negativebecause the pressure is decreasing.Yes, the rate of pressure change is getting closer to

zero.R4. a. See text definition of derivative.

b. Differentiate.c. ll y = xn, then !'= 1;n-1.d. S6e solution to'Problem Set 3-4, Problem 35'e. See the Proof in Section 3-4.

r. l] is pronounced "Dee Y, dee x."

d...ft(vl i" pronounced "Dee, dee x, of y'"

Both mean the derivative of y with respect to x'

g. i. f(x) =fY;et' + f'(x) =$x+rs

ii. g(x) = r-t -* - x +7 = g'(x) = -28rs-| -riii. h(x) = 73 + h'(x) = 0

The graph of the derivative is the same as the sinegrap[, tiut inverted in the y-direction. Thus, (cos x)'

= -sin x is confirmed.cos 1 = 0.540302305

Numerical derivative = 0.54030221The two are very close!Comoosite {unction.f'(x) = -2x sin (x2)

b.

c.

d.

42 Colculus: Concepts ond Applicotions Problem Set 3'10