A selection of problems from previous Chapter 3 tests. (Note, these problems are not intended as a
comprehensive review.)
dy
dx
♦
Find
. Show all work, simplify all answers:
1.
y = x 2 (6x ! 2) 2 . (Leave your answer in fully factored form.)
2.
y=
3.
y = x sec
4.
3y ! 2xy 2 + 2y 5 = y 2
5.
y = (2x 2 + 3x ) (7x ! 5)
6.
y=
tan 2x !1
sec 2x
7.
For
4
2
(5)
y = 7x ! 2x + x ! 13 find y .
8.
3
2
y = 5x (3x ! 4x + 3) (There’s a fast way or a long way. . .)
9.
y = (5x ! 7) (2x + 3)
1
1+ cot 2x
csc2 x
2
(There’s a fast way or a long way to do this. This hint is not given on test.)
( 2x )
3
(There’s a fast way or a long way to do this. This hint is not given on test.)
5
10. y =
2x
x
2
4
4
3
!1
11.
y = tan 2x csc 2x (There’s a fast way or a long way. . .)
12.
4
4
y = sin x ! cos x (Think simplification)
♦
Application problems from previous tests:
1.
Find the x-coordinates of all points on the curve
horizontal.
2.
Find the equations of both lines through the point (2,-3) that are tangent to the parabola
(Note: The point (2,-3) is not a point on the parabola.)
3.
Find the slope(s) of the line(s) tangent to the curve
4.
The radius of a spherical gas tank is 6.5 meters with a possible measurement error of 1cm . Use
3
4
differentials to estimate the error in the calculated volume of the sphere. (Volume of sphere = 3 !r ) Also
estimate the percent error. (From section 3.11)
5.
If
y = 2sin x + sin 2 x at which the tangent line is
y = x2 + x .
3xy ! 2x + y 2 = 16 at the point(s) when x = 1.
y = 5x 2 ! 7x + e x models the number of pages, y, read by Cabrillo students in x years, what are the
units of
dy
dx
?
♦
Related rate problems from previous tests:
1.
Sand is falling onto a conical pile at a constant rate of 10 m min such that the diameter of the pile is always
3 times the height. Find the rate at which the height of the pile is rising when the pile is 5m high. (Don't
forget the units in your answer!) What is happening to the rate of change of the height as the pile grows
larger? Justify your answer analytically by examining dh dt
2.
A water tank has the shape of an inverted circular cone. The tank has height 4m and diameter 4m. If water
m3
is being pumped into the tank at a rate of 2 min ,
3
a.
b.
!
Find the rate at which the water level is rising when the water is 3m deep.
What is happening to this rate as the water rises? Justify your answer mathematically.
3.
A stone dropped into a still pond causes a circular wave. Assume the radius of the wave expands at a
constant rate of 2 ft/s. How fast is the area expanding when the radius is 3 ft? What is happening to the
rate of change of the area as the circle grows larger? Justify your answer analytically by discussing dA dt .
4.
The angle of elevation of the sun is decreasing at a rate of 0.25 rad/hr. How fast is the shadow cast by a
400-ft-tall building increasing when the angle of elevation of the sun is ! 6 ?
♦
Proofs from previous tests:
1.
Prove
d
dx
(tan x ) = sec2 x
2.
Prove
d
dx
(sec x ) = sec x tan x
3.
Prove
sinh("x) = "sinh x
4.
Prove
cosh x is an even function
5.
Prove
cosh( x + y ) = cosh x cosh y + sinh x sinh y