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Math 5A
1) Letj(x) =
.
Some Practice Problems for Exam 2 (3.1-4.2)
for x 2: 4
ax+b
{ sin ( ~ x ) + 2
for x < 4
a) First find a and b so j(x) is differentiable at x = 4.
b)Now find/ex)
c) Now find!'(x)
2) Find the following limits. If the limit Does Not Exist fully explain why.
a)
lim
x->O
sin4x =
x
b)
lim
x->O
ta~3x =
Slnx
C)
Express
17
lim
x--+!
-
1 as a derivative and then evaluate.
-
e-X at x = 1.
X
x-I
3) Find the equation of the tangent line to the functionj(x) = 2x 2
4) For what values of r does the function y = en; satisfy fna.equation y" + 5y' - 6y = O? 5) A particle is moves along a horizontal line. Its distance away, S(t), from a fixed point is given by Set) = t 3 - 12t + 3, t 2: 0 in seconds and Set) in feet away from the fixed point. a) Find when the velocity and acceleration functions. b) Find when the particle is at rest. c) Find when the particle in moving the positive direction SOLN When v >0 f) When is the particle speeding up or slowing down? SOLN When a and v have the same sign When a and v have opposite signs
h) Find the distance travelled during the first 3 seconds.
SOLN find when v=O and use If-fl +If-fl
6) Calculate the following derivatives. Note, a, b, c are constants.
a)
..1L[ (at 2 -
c)
:fx
[e(Sin7x)2]
e)
ft
(Trl
g)
d~ [arccos3x + arcsin4x + arctan5x]
m
bt+ c)e l ]
d)
+ Tr11: + t11: + tl)
"'"
I) Ifj(x)
7)
-.fL ( cosx - eX )
b)
= xe X Find.f'(x).
~
rx
:10 ((tanO)4+ tan (48)+tan(04))
f):fx (In(x3) + log4(3x) + Inl 3x: 1 I)
h)-.fL ( (x + 1)5(2 + x)--6 x + )
dx
(x - 1) 7 JX+T
j) -.fL[(sinxY +xsinX]
~
Implicit Differentiation
a) Find : .
i)x 2y+.xy2=3x
iii)tan(~) =x+y
iv)l+x=exy2
b) Find the tangent line to the Folium of Descartes, x 3 + y3 = 6.xy, at the point (3,3).
c) At what pOints on the curve x 3 + y3 = 6yx is the tangent line(s) horizontal?
ii)4cos2xsin3y= 1
Related Rates
a) A balloon is rising vertically above a level, straight road at a constant rate of 1 ftlsec. Just when the
balloon is 65 ft above the ground, a bicycle passes under it, going 17 ftlsec. How fast is the distance
between the bike and the balloon increasing 3 seconds later?
8)
1
b) Water is being poured into a conical reservoir at the rate of n ft3 per second. The reservoir has a radius
of 6 feet across the top and a height of 12 feet. At what rate is the depth of the water increasing when the
depth is 6 feet?
c) A fish is reeled in at a rate of 1 foot per second from a point 10 feet above the water. At what rate is the
angle between the line and the water changing when there is a total of 25 feet of line out?
d) An observer is tracking a plane flying at an altitude of 5000 ft. The plane flies directly over the observer
on a horizontal path at the fixed rate of 1000 fUmin. Find the rate of change of the distance from the plane
to the observer when the plane has flown 12,000 ft. after passing directly over the observer.
9) Use the graph below to find the derivatives or state it does not exist. The graph, of j(x) is the thin
dotted line, and the graph of g(x) is solid thicker line.
a) LetP(x)
= j(x) * g(x). Find pI(I) and p I(2).
b) LetK(x)
= 1c~. Find IC(3)
c) Let U(x)
= fig(x».
Find U(3)
10) Linear Approximations and Differentials a) i) Find the linear approximation for j(x) = e-5x at x = O.
ii) Approximate e--{),O]. b) i) Find the linearization of the functionj(x) = Jx + 3 at x = 1. ii) Use the linearization to approximate J3.95 and J4.04. iii)Are these approximations overestimates or underestimates? c) Evaluate dy if y = x 3 - 2x 2 + 1, x = 2, and dx = 0.2.
d) The radius of a cylinder with a fixed height of 10 cm is given as 40 cm with a maximum error in
measurement of 0.2 cm.
i) Use differentials to estimate the maximum error in the calculated volume and surface area of the
cylinder.
ii) What is the relative error?
11) Extrema(Max and Min) Problems a) Find the critical numbers and absolute extrema for j(x) = (2x - 1)(3 - x)3 on- the interval [1,4]. b) Find the critical numbers and absolute extrema for j(x) = -sinx + +x on the interval [0,3nJ. i
c) Find the critical numbers and absolute extrema for fix) = - JX +
on the interval [.01,9]. x
d) Letj(x) = (2x+ I)e- • Find wherej(x) has absolute extrema on the interval [-1,4]. e) Find the critical numbers and absolute extrema forj(x) = x~-4 on the interval [-4,4].
x+4
12) State the Mean Theorem Value and Rolle's Theorem
a) Find the c satisfying the conclusion of the Mean Value Theorem for j(x) = -2x 3 + 3x 2 on the interval
[-2,1]
b) If possible apply Rolle's Theorem to j(x) = (-x + 3 )3x on the interval [0,3J
c) Letj(x)
=
~~
f . Show that there is no value c satisfying the conclusion of the MVT on the interval
[-2,2] . Why does this not contradict the MVT?
13) Derive Derivatives of Hyperbolic Functions
2