Section 2.3
Differentiation formulas: If f is differentiable at x and if c and n are real numbers, then
(1)
(3)
d
dx
c=0
d
dx
cf (x) = cf 0(x)
(2)
d
dx
(4)
d
[f (x) dx
Exercise.
xn = xn¡1
g(x)] = f 0(x) g 0(x)
1. Find the derivatives of following functions.
(1) y = 2
(2) y = x5
(4) f(x) =
x7 + 2x5 + 4x4 ¡ 3x3
x3
(7) y = (x3 ¡ 4x2)(3x4 ¡ 2x)
(10) f (x) =
x3 ¡ 2x2 ¡ x + 3
p
x
x2 ¡ 2a x + a2
(13) y =
,
x¡a
a is a constant and x =
/a
(3) f(x) = 4x5 + 3x2 ¡ 7
(5) y = e
(6) y = (x ¡ 1)(3x5)
(8) f (x) = 3x3 + 3
(9) y = 7
p
(11) y = x (x + 1)
(12) f (x) = 3x 2 ¡ 4x 3 + x
p
5 3
x
1
2
¡2
5
¡ 7xe + x
(14) y = ax2 + bx + c,
a; b; c are constants
2. Find the equation of the tangent line to the curve y = ex at x = ln4.
3. If f (3) = 4, g(3) = 2, f 0(3) = 5, and g 0(3) = 7, find (a) 6 f 0(3) and (b) (f + g) 0(3).
1
4. Find the third derivative (denoted f (3)) of f (x) = 3x3 + 4x 2 ¡ x
5. Find the nth derivative (denoted f (n) or
dn
)
dxn
¡2
3
.
of f(x) = 5x3 + 3x ¡ 2.
6. Write the equation of the tangent line to the curve y = 3x2 ¡ x that is parallel to the line 2x ¡ y = 11.
7. Suppose the derivative of the function f exists and f(3) = 1, f 0(3) = 4. Let g(x) = x2 + f (x) and
h(x) = 3f (x).
a) Find an equation of the line tangent to y = g(x) at x = 3.
b) Find an equation of the line tangent to y = h(x) at x = 3.
Section 2.3 (cont)
Differentiation formulas:
d
(f
dx
g(x)) = f 0(x)g(x) + f(x)g 0(x).
d f
f 0(x)g(x) ¡ f (x)g 0(x)
2. If f and g are differentiable at x and g(x) =
/ 0, then dx g (x) =
.
[g(x)]2
1. If f and g are differentiable at x, then
3. If n is any real number, then
d
(xn) = nxn¡1.
dx
Exercise.
1. Find the derivatives of following functions.
(1) y = x5
(4) y =
p
(5) y = 7x4
x3 + 2x2 + x ¡ 3
x
p
(7) s(t) = 4(t ¡ 2) t
(10) y =
p
(3) f (x) = x5
(2) y = x3
(6) y = (x3 ¡ 4x2)(3x5 + 2x4 ¡ 3x)
x3 +
p
x
(8) y = (x2 + 3x)(4 ¡ 2x)
(9) f (x) = 3x2 ¡ 5x
(x ¡ 2)(x + 4)
2x
p
(11) y = (2 3 x ¡ 1 )(x + 1)¡1
(12) f(x) = 3x3 ¡ 5x
a x2 ¡ 2bx + c 2
(14) h(x) =
(13) y =
,
x¡a
where a; b; c are constants
and x =
/ a.
x4 +
p
x
xf (x)
,
g(x)
where f ; g are differentiable
functions and g(x) =
/ 0.
2x2
2. Find the equation of the tangent line to the curve y = 3x ¡ 1 at x = 1.
3. If f (3) = 4, g(3) = 2, f 0(3) = 5, and g 0(3) = 7, find (a)
4. If f (x) =
p
0
f
(3) and (b) (fg) 0(3).
g
x g(x) where g(8) = 6 and g 0(8) = ¡3, find f 0(8).
x¡1
5. Write the equations of all tangent lines (in slope intercept form) to y = x + 1 that are parallel to the
line x ¡ 2y = 3.
6. Suppose the line tangent to the graph of f at x = 2 is y = 4x + 1 and suppose the line tangent to the
graph of g at x = 2 is y = 3x ¡ 2. Find the line tangent to the curve y = f (x)g(x) at x = 2.
7. Discuss notation:
h
d xf (x)
dx g(x)
i
x=4
.
p
x
8. Find the equation of the normal line to the curve y = x + 1 at x = 4.
Section 2.4
sin !0 Limit formulas for trignometric functions: lim
Exercise.
Find the limits.
1 ¡ sin 7
(1) lim
!0
sin 5
x!0 8x
(2) lim
tan 2
!0 sin (8) lim
sin(x ¡ 1)
x!0
!0
cos ¡ 1
= 0.
!0
(5) lim x2 + x ¡ 2
(7) lim
!0
(3) lim
1 ¡ cos2 4t
t2
t!0
(4) lim
= 1, lim
sec ¡ 1
cos ¡ 1
sin sin(x ¡ 1)
(6) lim x2 + x ¡ 2
x!1
(9) lim
x!0
tan ax
,
bx
a; b are constants and b =
/0
Derivative formulas for trignometric functions:
d
dx
sin x = cos x
d
cos x = ¡sin x
dx
d
tan x = sec2x
dx
d
dx
cot x = ¡csc2x
d
sec x = sec x tan x
dx
d
csc x = ¡csc x cot x
dx
Exercise.
1. Find the derivatives of following functions:
(1) f (x) = x sin x
(2) f(x) = 3 csc x ¡ 5 cos x
(4) y = (tan x)(x + sec s)
(5) y =
tan x + 3 cos x
1 ¡ 2x
1 + sin x
(3) f (x) = x + cos x
2. Find the equation of the normal line to the curve y = sec x ¡ 2 cos x at the point
cos x
x csc x
(6) y = 4 cot x
¡
3
3. Find the points on f (x) = 2 + sin x at which the tangent line is horizontal on [0; 2).
4. Given sin(x + y) = sin x cos y + cos x sin y, use the definition of derivative to prove
2
5. Find y 0 for y = 3x4 ¡ 2x¡1 + x 3 + e4x.
;1 .
d
dx
sin x = cos x.
Section 2.5
Recall the differentiation formulas:
d
dx
c=0
d
dx
d
dx
ekx = kex
d
(f
dx
Generalization of
when q is even.
Exercise. Find
d
[f(x) dx
cf(x) = cf 0(x)
d
dx
d
dx
g(x)) = f 0(x)g(x) + f (x)g 0(x)
f
(x)
g
d
dx
g(x)] = f 0(x) g 0(x)
=
f 0(x)g(x) ¡ f (x)g 0(x)
[g(x)]2
xn = xn¡1: if p and q are integers and q =
/ 0, then
d
dx
p
p
xq = q x
xn = xn¡1
if g(x) =
/0
p
¡1
q
, provided x > 0
3
d
(x3 + 4x5) 5 .
dx
To differentiate composite functions, we apply the Chain Rule: If F (x) = f (g(x)) with g differentiable
at x and f differentiable at g(x); then
F 0(x) = f 0(g(x)) g 0(x)
Alternative notation: if y = f (u) and u = g(x), then
dy
dx
dy du
dx
= du
Example. Derivative formulas where u is a function in x:
d
dx
un = nun¡1 dx
du
d
dx
[g(x)]n = n[g(x)]n¡1 g 0(x)
d
dx
sin u = cos u dx
du
d
dx
cos u = ¡sin u dx
d
dx
cot u = ¡csc2 u dx
d
dx
sec u = sec u tan u dx
du
du
du
d
dx
tan u = sec2 u dx
du
d
dx
csc u = ¡csc u cot u
du
dx
Exercise.
1. Find the first derivative of the following functions by using the chain rule:
p
(1) y = (7 ¡ 3x)10
(2) f(x) = 3 1 + tan 2x
(3) g(x) = 4 sec (5x)
(4) y = sin (e3x)
(5) y = sin2 x + sin (x2)
(6) f (x) = p
(7) y = (2x ¡ 1)2 ecsc3x
(8) f(x) = cos2 (cot (3x))
(9) y =
(10) y = f (g(cot (x)))
(11) f (x) =
2. Let f (x) =
x3 + 1
x3 ¡ 1
2
3
p
x2 ¡ 1 . Find the domain of f and f 0 in interval form.
3x
x2 ¡ x
p
f (x)g(x)
1
(12) f (x) =
(3x + 1) 3
1
(1 ¡ 4x) 2
Section 2.6
dy
Implicit differentiation: Regard y as a function implicitly depends on x, and solve for y 0 = dx from the
given equation by differentiating both sides of the equation with respect to x.
Exercise.
dy
1. Find y 0 = dx for each of the following.
(1) x2 + y 2 = 7
(2) x2 ¡ 2xy + y 3 = e
(4) x cos (xy) = 1
(5) tan (x ¡ y) = x4y
p
(3) 1 + 2x2y 2 = 2y
2. Write the equation of the tangent line to the curve x2y 2 = (y + 1)2(4 ¡ y 2) at (0; ¡2).
3. Write the equation of the normal line to the curve x2y2 = (y + 1)2(4 ¡ y 2) at (0; ¡2).
d dy
dx
Second derivative of y = f (x) is y 00 = dx
Exercise. Find y 00 for sin x + x2y = 9.
d2y
= dx2 .
Definition. (Orthogonal curves) Two curves are orthogonal if the tangent lines are perpendicular at each
point of intersection.
Two families of curves are orthogonal trafectories if each curve in one family is orthogonal to each curve
in the other family of curves.
Exercise.
1. Show that 2x2 + y 2 = 3 and x = y 2 are orthogonal.
2. Show that the family of curves x2 + y 2 = ax and x2 + y 2 = by are orthogonal trajectories.
3. Find the values of c so that the curves cx2 + y 2 = 3 and x = y 2 intersect orthogonally at (1; 1).
Section 2.7
Definition. (Average velocity and Instantaneous velocity) Let s = f(t) be the position function of
an object moving along a line.
The average velocity of the object over the time interval [a; a + t] is the slope of the secant line between
f (a + t) ¡ f (a)
(a; f (a)) and (a + t; f(a + t)):
.
t
The instantaneous velocity at t = a is the slope of the line tangent to the position curve, which is the
f (a + t) ¡ f (a)
derivative of the position function: v(a) = lim
= f 0(a).
t
t!0
Definition. (Velocity, Speed, and Acceleration) Suppose an object moves along a line with position
s = f (t). Then,
ds
Velocity at time t: v = dt = f 0(t).
Speed at time t: jv j = jf 0(t)j.
Acceleration at time t: a =
dv
dt
=
d2 s
dt2
= f 00(t).
Exercise.
1. Suppose the position of an object moving horizontally after t seconds is given by the function f (t) =
¡6t3 + 36t2 ¡ 54t, 0 6 t 6 4, where s is measured in feet, with s > 0 corresponding to positions right
of the origin.
a) Graph the position function.
b) Find and graph the velocity function. When is the object stationary, moving to the right, and
moving to the left?
c) Determine the velocity and acceleration of the object at t = 1.
d) Determine the acceleration of the object when its velocity is zero.
2. Suppose a stone is thrown vertically upward from the edge of a cliff on Mars (where the acceleration
of gravity is only about 12 ft/sec2) with an initial velocity of 64 ft/sec from a height of 192 feet above
the ground. The height s of the stone above the ground after t seconds is given by s = ¡6t2 + 64t + 192.
a) Determine the velocity v of the stone after t seconds.
b) When does the stone reach its highest point?
c) What is the height of the stone at its highest point?
d) When does the stone strike the ground?
e) With what velocity does the stone strike the ground?
Section 2.8
In a related rate problem, the idea is to compute the rate of change of one quantity in terms of the rate of
change of another quantity.
Strategies for solving related rates problems:
1. Read the problem carefully.
2. Draw a diagram if possible.
3. Introduce notation. Assign symbols to all quantities that are functions of time.
4. Express the given information and the required rate in terms of derivatives.
5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry
of the situation to eliminate one of the variables by substitution.
6. Use the Chain Rule to differentiate both sides of the equation with respect to t.
7. Substitute the given information into the resulting equation and solve for the unknown rate.
Following problems are from the textbook.
Exercise.
1. The volume of a cube decreases at a rate of 0.5 ft3 per minute. What is the rate of change of the side
length when the side lengths are 12 feet?
2. A spherical balloon is inflated and its volume increases at a rate of 15 in3 per minute. What is the
rate of change of its radius when the radius is 20 inches?
3. A piston is seated at the top of a cylindrical chamber with radius 5 centimeters when it starts moving
into the chamber at a constant speed of 3 cm / s. What is the rate of change of the volume of the
cylinder when the piston is 2 cm from the base of the chamber?
4. A jet ascends at a 10 degree angle from the horizontal with an airspeed of 550 mph (its speed along
its line of flight is 550 mph). How fast is the altitude of the jet increasing? If the sun is directly
overhead, how fast is the shadow of the jest moving on the ground?
5. A 13-ft ladder is leaning against a vertical wall. If the foot of the ladder is puuled away from the wall
at a rate of 0.5 ft/s, how fast is the topofthe ladder sliding down the wall when the foot of the ladder
is 5 feet from the wall?
6. Sand falls from an overhead bin and accumulates in a conical pile with a radius that is always three
times its height. Suppose the height of the pile increases at a rate of 2 cm/s when the pile is 12 cm
high. At what rate is the sand leaving the bin at that instant?
7. At what rate is soda being drunk out of a cylindrical glass that is 6 inches tall and has a radius of 2
inches? The depth of the soda decreases at a constant rate of 0.25 in/s.
8. An inverted conical water tank with a height of 12 feet and radius of 6 feet is drained through a hole
in the vertex at a rate of 2 ft3 /s. What is the rate of change of the water depth when the water depth
is 3 feet? (Hint: use similar triangles)
9. A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min . What
is the rate of change of the area at hte instant the radius is 10 ft?
Exercise. (continued)
10. A spherical snowball melts at a rate proportional to its surface area. Shot that the rate of change of
the radius is constant. (Hint: surface area = 4r2)
11. A surface ship is moving in a straight line at 10 km / hr. At the same time, an enemy subnarine
maintains a position directly below the ship while diving at an angle that is 20 degress below the
horizontal. How fast is the submarine's altitude decreasing?
12. An observer is 20 meters above the ground floor of a large hotel atrium looking at a glass-enclosed
elevator shaft that is 20 meters horizontally from the observer. The angle of elevation of the elevator
is the angle that the observer's line sight makes with the horizontal (it may be positive or negative).
Assuming that the elevator rises at a rate of 5 m/s, what is the rate of change of the angle of elevation
when the elevator is 10 meters above the ground? When the elevator is 40 meters above the ground?
13. The bottom of a large theater screen is 3 feet above your eye level and the top of the screen is 10 feet
above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate
of 3 ft/s while looking at the screen. What is the rate of change of the viewing angle , when you
are 30 feet from the wall on which the screen hangs, assuming the floor is flat?
14. A camera is set up at the starting line of a drag race 50 feet from a dragster at the starting line. Two
seconds after the start of the race, the dragster has traveled 100 feet and the camera is turning at a
rate of 0.75 rad/s while filming the dragster.
a) What is the speed of the dragster at that point?
b) A second camera filming the dragster is located on the starting line 100 feet away from the
dragster at the start of the race. How fast is this camera turning 2 seconds after the start of
the race?
p
15. A particle moves along the curve y = x in such a way that the y-value is increasing at a rate of 2
units/s. At what rate is the x-value changing when x = 1?
16. A ladder 20 feet long leans against hte wall of the side of a building. If the bottom of the ladder
slides away from the building horizontally at a rate of 2 ft/s, how fast is the ladder sliding down the
building when the top of the ladder is 12 feet above the ground?
17. A girl starts at a ooint A and runs east at a rate of 10 ft/s. One minute later, another girl starts at
A and runs north at a rate of 8 ft/s. At what rate is the distance between them changing 1 minute
after the second girl starts running?
18. A spherical balloon filled with gas has a leak that permits the gas to escapte at a rate of 15 m3 per
minute. How fast is the surface area of the balloon shrinking when the radius is 4 meters?
19. Water is flowing into a vertical cylindrical tank of diameter 6 meters at the rate of 5 m3 per minute.
Find the rate at which the depth of water is rising.
20. Consider a container in the form of a right circular cone (vertex down) with radius of 4 meters and
height of 18 meters. If water is poured into the container at the constant rate of 16 m3 per minute,
how fast is the water level rising when the water is 8 meters deep?
Exercise. (continued)
21. A man 6 feet tall is walking at the rate of 3 ft/s toward a streetlight 18 feet high.
a) At what rate is his shadow length changing?
b) How fast is the top of this shadow moving?
22. A man starts walking north at 4 ft/s from a point P . Five minutes latera woman starts walking south
at 5 ft/s from a point 500 feet due east of P . At what rate are the people moving apart 15 minutes
after the woman starts walking?
23. The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing
at a rate of 2 cm2 /min . At what rate is the base of the triangle changing when the altitude is 10 cm
and the area is 100 cm2?
24. A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 m
wide at the bottom, 80 cm at the top, and has height 50 m. If the trough is being filled with water
at the rate of 0.2 m3 /min , how fast is the water level rising when the water is 30 m deep?
25. Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate
of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between
the sides of fixed length is 3 .
26. Suppose x is a differentiable of t and suppose that when x = 15,
dx
dt
= 3. Find
dy
dt
if y 2 = 625 ¡ x2.
Section 2.9
Definition. (Linear approximation) Suppose f is differentiable on the interval I which contains point a.
The linearization of f at a is the linear function L(x) = f (a) + f 0(a)(x ¡ a). The approximation f (x) L(x),
or f(x) f (a) + f 0(a)(x ¡ a) is called the linear approximation of f at a.
We have already seen that a curve lies very close to its tangent line near the point of tangency. This is
the basis for a method of finding approximate values of functions. We use the tangent line at (a; f (a)) as
an approximation to the curve y = f (x) when x is near a.
y ¡ y1 = m(x ¡ x1)
y ¡ f(a) = f 0(a)(x ¡ a)
y = f (a) + f 0(a)(x ¡ a)
Exercise.
1. Find the linear approximation and estimate the following:
(a) f (x) = 8 + x2 at a = 1, estimate f (1.03)
(b) f(x) = cos x
at a = 3 , estimate f (1.1)
p
p
p
2. Find the linearization of f (x) = x + 3 at a = 1 and use it to approximate 3.98 and 4.05 .
3. Use linear approximation to estimate
p
55 .
Section 3.1
Definitions:
1. Absolute Maximum and Minimum
Let f be a function defined on an interval I containing the point c. Then f has an absolute (or
global) maximum on I at c if f (c) > f (x) for every x 2 I. Similarly, the function f has an absolute
(or global) minimum on I at c if f (c) 6 f(x) for every x 2 I.
2. Local Maximum and Minimum
Let f be a function defined on an interval I and c an interior point in I, i.e., there exists an open
interval (a; b) such that c 2 (a; b) I. If f (c) > f (x) for all x in some open interval containing c, then
f(c) is a local maximum value of f . If f (c) 6 f (x) for all x in some open interval containing c, then
f(c) is a local minimum value of f .
3. Critical Point
An interior point c of the domain of f at which f 0(c) = 0 or f 0(c) fails to exist is called a critical
point of f .
Theorems:
1. Extreme Value Theorem
A function that is continuous on a closed interval has an absolute maximum value and an absolute
minimum value on that interval.
2. Fermat's Theorem
If f has a local maximum or minimum at c and f 0(c) exists, then f 0(c) = 0.
Method of finding absolute extrema of f on a closed interval [a; b]:
1. Find the critical points of f in [a; b]:
2. Evaluate f at each endpoint and each critical point in the domain.
3. The largest value of f is the absolute maximum and the smallest value of f is the absolute minimum.
Exercise.
1. Find the critical points for each of the following:
(1) f (x) = x3 + 7x2 ¡ 5x
(2) f (x) = xe¡1/2
(4) f (x) =
p
x2 ¡ 9
(7) g(t) = cos2 t
on [0; 2]
(3) f (x) = x7/3 + x4/3 ¡ 3x1/3
p
(5) f (x) = 3 x2 ¡ 9
(8) f (x) = sin2 3x
(6) f () = 4 ¡ tan on [0; 2]
on [0; 2]
p
(9) f (x) = x3 x2 ¡ 9
2. Find the absolute extrema and sketch the graph for each of the following:
(1) f (x) = x3 + 7x2 ¡ 5x on [¡1; 6]
(2) f (x) = x7/3 + x4/3 ¡ 3x1/3 on [¡1; 1]
(3) f (x) = x4 + 4x3 ¡ 2x2 ¡ 12x
(5) f (x) = sin3 x on [0; ]
on [0; 2]
(4) f (x) = xe¡1/2
on [0; 5]
Section 3.2
Theorems:
1. Rolle's Theorem
Let f be a function that satisfies the following three hypotheses:
a) f is continuous on [a; b];
b) f is differentiable on (a; b),
c) f(a) = f (b).
Then there is a number c in (a; b) such that f 0(c) = 0.
2. Mean Value Theorem
Let f be a function that satisfies the following:
a) f is continuous on [a; b];
b) f is differentiable on (a; b).
Then there is a number c in (a; b) such that f 0(c) =
f (b) ¡ f (a)
,
b¡a
or, f 0(c)(b ¡ a) = f(b) ¡ f(a).
3. Theorem 3.2.5
Constantly zero derivative implies constant function: If f is differentiable and f 0(x) = 0 at all
points of an interval I, then f is a constant function on I.
4. Theorem 3.2.7
Functions with equal derivatives differ by a constant.
Exercise. Show that the function satisfies the hypotheses of Rolle's Theorem and find all c that satisfies
the conclusion of Rolle's Theorem:
p
(1) f(x) = x x + 6 on [¡6; 0]
(2) f (x) = (x ¡ 1)¡2 on [0; 2]
(3) f(x) = x2 ¡ 4x + 1
(5) f(x) = sin 2x
on [0; 4]
(4) f (x) = x3 ¡ 4x2 + 2x + 5
on [0; 2]
on [¡1; 1]
Exercise. Show that the function satisfies the hypotheses of the Mean Value Theorem and find all c that
satisfies the conclusion of the theorem:
x
(1) f (x) = x + 2
on [1; 4]
(3) f (x) = 3x2 + 2x + 1
(5) f (x) = x3 + x ¡ 1
on [¡1; 1]
on [0; 2]
2x
(2) f (x) = x + 1
p
(4) f (x) = 3 x
(9) f (x) =
on [1; 3]
on [0; 1]
p
x2 + x + 4
on [0; 3]