MATH 151 Engineering Math I, Spring 2014
JD Kim
Week7 Section 3.8, 3.9, 3.10
Section 3.8 Higher Derivatives
Definition If y = f (x), then the second derivative of f (x) is the derivative of the
first derivative. We denote the second derivative as y ′′ = (f ′ (x))′ = f ′′ (x). Similarly,
the third derivative is the derivative of the second derivative, denoted by f ′′′ (x). In
general, the nth derivative of f (x) is denoted by f (n) (x).
Ex1) Find the second derivative of f (x) = x cos x.
Ex2) Find the f (10) (x), for f (x) = cos(2x).
1
Ex3) Find a general formula for nth derivative for f (x) =
1
.
x
Acceleration If s = s(t) is the position function of an object that moves in
a straight line, the first derivative represents the velocity v(t) of the object as a
function of time:
ds
v(t) = s′ (t) =
dt
The instantaneous rate of change of velocity with respect to time is called the acceleration a(t) of the object:
a(t) = v ′ (t) = s′′ (t)
or, in Leibniz notation,
a=
dv
d2 s
= 2
dt
dt
Ex4) If s(t) = 2t3 − 7t2 + 4t + 1 is the position of a moving object at time t, where
s(t) is measured in feet and t is measured in seconds, find;
4-1) The velocity at time t.
2
4-2) The acceleration at the times when the velocity is zero.
t
Ex5) If r(t) =< , t2 >
2
5-1) Sketch the curve.
5-2) Plot the position, tangent and acceleration vectors at the point corresponding
to t = 2.
3
Ex6) Find f ′′ (x) if f (x) = g(x3 ) + (g(x))3 .
Ex7) Find y ′′ by implicit differentiation for the equation x2 + 6xy 2 = 8.
4
Section 3.9 Slopes and Tangents of Parametric
Curves.
Derivatives of Parametric Curves
dy
dy
If x = f (t) and y = g(t), then
= dt . This gives us a way to find the slope of
dx
dx
dt
dy the tangent line to the parametric curve at t = t0 : m =
.
dx t=t0
Ex8) Find
√
dy
if x = (3t − 1)2 and y = t t.
dx
Ex9) If x = 1 − t3 and y = t2 − 3t + 1, find an equation of the tangent line
corresponding to t = 2.
5
Ex10) If x = 2t + 3 and y = t2 + 2t, find the equation of the tangent line at the
point (5, 3).
Ex11) If x = t3 − 3t2 and y = t3 − 3t, find all points on the curve where the
tangent line is vertical or horizontal.
6
Ex12) Show the curve x = cos t and y = (sin t)(cos t) has two tangents at (0, 0).
Find the equations of these tangent lines.
Ex13) At what points on the curve x = t3 + 4t, y = 6t2 is the tangent line parallel
to the line with equations x = −7t, y = 12t − 5?
7
Section 3.10 Related Rates
In a related rates problem, the idea is to compute the rate of change of one
quantity in terms of the rate of change of another quantity (which may be more easily
measured). The procedure is to find an equation that relates the two quantities and
then use the Chain Rule to differentiate both sides with respect to time.
Related Rates
In this section, we have two or more quantities that are changing with respect
to time. Our first goal is to find a formula that relates the quantities. Then we will
solve for the desired rate of change.
dx
If x is changing with respect to time t, then x = x(t) and
is the rate of change x
dt
with respect to time.
Ex14) Suppose both x and y are changing size with respect to time t (assume x
and y are both positive). At the instant when x = 4, the rate of change of x with
respect to t is −2. If x and y are related by the equation x3 + 4y 2 = 84, find the
corresponding change in y at this same instant.
8
Ex15) A stone is thrown into a still pond and circular ripples move out. The
radius of the disturbed region increases at a rate of 3 f t/sec. Find the rate at which
the area of the disturbed region is increasing when the farthest ripple is 20 f t from
the place at which the rock struck the pond.
9
Ex16) A storm is 50 miles offshore and its path is perpendicular to a straight
shoreline. It is approaching the shore at a rate of 4 mph. A van traveling along
the shoreline wants to stay exactly 50 miles from the storm and remain along the
shoreline. The van starts at the point on the shoreline in the path of the storm.
16-1) Determine formula for the speed that the van must maintain to remain
50 miles from the storm.
16-2) Find the speed of the van when the storm is 40 miles from the shore.
10
Ex17) A water tank has the shape of an inverted circular cone with radius
5 meters and height 7 meters. A solution is being poured into the cone in such
a way that the height of the fluid is increasing at a rate of 1/2 meter per minute. At
what rate is the volume increasing when the height is 4 meters?
11
Ex18) A ladder 10 feet long rests against a vertical wall. If the bottom of the
ladder slides away from the wall at a speed of 2 f eet/s, how fast is the angle between
the top of the ladder and the wall changing when the angle is π/4 radian?
12
Ex19) The length of a rectangle is increasing at the rate of 2 f eet per second,
while the width is increasing at the rate of 1 f oot per second. How fast is the area
increasing when the length is 5 f eet and the area is 50 squaref eet?
13
Ex20) If a snowball melts so that its surface area decreases at a rate of 1 cm2 /min,
find the rate at which the diameter decreases when the diameter is 10 cm.
14
Ex21) At noon, ship A is 100 km west of ship B. Ship A is sailing south at
35 km/h. Ship B is sailing north at 25 km/h. How fast is the distance between them
changing at 4 : 00pm?
15
Ex22) A trough is 10 f eet long and its ends have the shape of isoceles triangles
that are 3 f eet across the top and have a height of 1 f oot. If the trough is filled with
water at a rate of 12 cubic f eet per minute, how fast is the water level rising when
the water is 6 inches deep?
16
Ex23) Two sides of triangle have length 5 m and 4 m. The angle between them
π
rad/s. Find the rate at which the area of the triangle is
is increasing at a rate of
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increasing when the angle between the sides of fixed length is π/6.
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2
2
Ex24)
√
√ A point moves around the circle x + y = 9. When the point is at
(− 3, 6), its x coordinate is increasing at a rate of 20 units per second. How fast
is its y coordinate changing at that instant?
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Ex25) Water is leaking out of an inverted conical tank at a rate of 20 cubic inches
per second at the same time that water is being pumped into the tank at a constant
rate. The tank has height 6 feet and the diameter at the top is 4 feet. If the water
level is rising at a rate of 20 inches per second when the height of the water is 2 feet,
find the rate at which water is being pumped into the tank.
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