IAP GalculusGhapter 3 Test -AB -V2

Differentiate each of the following functions:

1) !=3xs-E*!

[sxQ - *" l t

Name:

2) y=log+(Zxz-x)

l \rAU - CZrl^^) .A'nq

4x-\

t {x- l

ln*

Y : cot( sin x)

St= -.lc'(:unx) cctl>.x

G""^,a)

4)3)

-.l

. l

5) What is the slope of the line tangent to the curve / = tan-1 4x atthe point at which1^^.

* = ;? Show your work.

(A) -z

(D)12

(B)

(E)

.t

L2

2

(c) 0

$--Lt + (.en)'

t \q

---

\ t tb Clq)"

q- ---=

t *,, C' l u)

- ,* =\=^

x=\/+

6) Use the following table of values to calculate the derivative of the given function at

x--2.

x f(x) a(x) f ' (x) a'(x\2 5 4 -3 94 3 2 -2 J

a Qf -lv

-u3

b) f(sQx))

h {'t3 (zx)) . 3'

'27') -}

f ' (3 tz 'z)) ' 3 'A' ' ) '7

?f ' (3C'r)) '3 ' (4) 'a

= { ' tz) Q 3'7-5U7' ,7

=- !S-\q

7) The position of a particle moving on the x-axis at time f > 0 seconds is: x(f) - et - fi

feet.(a) Find the average velocity of the particle over the interval [0,4]'

eu -G - (eo -'6)

?

'|,-- ̂ trn P

7/q-0

aq -L-\Lt

u4 -3= -- +

(b) Assuming that movement to the right is positive, il yhat direction (left or right)

and how fast is the particle moving at t = 1 seconds?

, \ +. \t )."1'v[€)= e- - IL ' '

t8) What is the instantaneous rate of change at x = 3 of the function / given by

=eL

=4

_L l . *t^\nr"t,"

T^G,A

\

is Pr\$'^ ,5o l-

(B)1B

(E) 13t6

I t- \a@z'9

+?' llo

wl+zg-c! - -+

-23=l6t

I f x3 + Zx'y - 4y = 7, then when x = L, y:ax

(B) _B

@-zat' h vz(* d& * s-zx =Q

7F' t Zr"t +ul

4t5 -tA d+ =eclp

\

e)

n\

b:-3 \ 'Y

?

h)-r*

(A) Z2

(D) 0

(c) -3

Zr.'40x

- Lt djl. c[/

di1dx-

4= - 3x'?r nv

L{-4.t

-3-+c-

@\

l *=\' b --- )

} +4VL

- .l_

=T-L

o>-\

10) The twice-differentiable function/is defined for all real numbers and satisfies thefollowing conditions:

/(o) = 2, f '(o) - -Q and /"(o) : 3

The function g is given by g(x) = eo* + f (x) for all real numbers, where a is aconstant. Find g'(0) and g" (0) in terms of a.

t-t--q

' ((r) = Q>o

=A'+

5!

c

' (n7= ^fr

+- f ' ( r)

g" (-0 =

g" [o)

a, e-o* t f t' [r)

=*lxf" r0)=I*9

-t'' J"t

,( _.t

1 1) The graph of the velocity v(t), in feet per second, of a bicycle racing on a straight road,

for 0 < t < 60, is shoivn. Also, given is a table of values for v(r), at 5 second intervals

of time r.

60

<n

40

30

v0

t0

(a) During what intervals of time is the acceleration of the bike positive? Give a

reason for your answer.

tr*

t(seconds)

v(r) (feetper

second)0 0) 1310 t615 2I20 2525 3530 4035 Aa

+L

40 4445 4850 4555 4460 i1

+l

0,'tr) u ts, f)Vl,' Wt( V Ls (tnIl'-

(b) Find the averase accelerstion of the bike, in ft/r, ou., the interval 0 < f < 60.

q? -o qke- = r--

60 -o bO

(c) Find one approximation for the acceleration of the bike, in rc/rr, utt = 30. Show

the computations you used to arrive at your answer.

Y'-T" =,q -71

vto

'l

EXTRA CREDIT

Find dydx

0/,r1^! = P'nt (o*)" n

A'n,1 = Qfr -Q',ne*

%!=Q.8'6Q.ane

h3=xax.I

i {1= x'ex n*/F 'i, + e'

* = xe* +ex

4 = q ' (*u*n"n)N)a

=[nd" tou" t. ')

= ft1*)'"' e" (x +l)

=(.")^on t'

(o f ) J

r. \ d-rar

I3