AP Calculus I
AP Calculus I
Notes 3.1
The Derivative and the Tangent Line Problem
Essentially, the problem of finding the tangent line at a point
P boils down to the problem of finding the slope of the tangent
line at point P. You can approximate this slope using the secant
line (not the trig):
Slope of the Secant line:
This is also known as the
______________________________________.
What if we wanted to find the slope of the line at a single
point, instead of two? Let’s try to find a method that can tell us
the slope at any single point using the slope formula:
Definition of Tangent Line with Slope m:
If is defined on an open interval containing c, and if the limit
exists, then the line passing through the point with slope m is the
tangent line to the graph of at the point .
The slope of the tangent line to the graph of at the point is
also called the slope of the graph of at .
Ex. 1:Find the slope of the graph of at the point .
Why does this answer make sense?
Ex. 2:Find the slopes of the tangent lines to the graph of at
the points and .
The Derivative of a Function
The limit used to define the slope of a tangent line is also
used to define one of the two fundamental operations of calculus –
DIFFERENTIATION
Definition of the Derivative of a Function
The derivative of at is given by
provided the limit exists
The process of finding the derivative of a function is called
differentiation. A function is differentiable at if its derivative
exists at and differentiable on an open interval if it is
differentiable at every point in that interval
.
Notations used to Denote the Derivative of
1. 2. 3. 4.
The notation is read as “the derivative of with respect to
.”
Ex. 3:Find the derivative of
This means that….
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
Ex. 4: Find the derivative of . Then find the slope of the graph
of at the points and . Discuss the behavior of at .
Ex. 5: Find the derivative with respect to for the function
.
What is the equation of the tangent line at ?
Differentiability and Continuity
Another alternate limit form of the derivative is: provided this
limit exists
The existence of the limit in this alternative form requires
that the one-sided limits of:
and
are equal. These one-sided limits are called the derivatives
from the left and from the right.
We say that is differentiable on the closed interval if it is
differentiable on and if the derivative from the right of and the
left of both exist.
Illustration
Sketch a graph of each function. Determine the intervals in
which the function is continuous and the intervals in which the
functions are differentiable.
1.
2.
3.
4.
Theorem – If is _____________________ at , then is
____________________ at
Summary:
1)
If a function is ___________________ at , then it is
____________________ at . Thus differentiability implies
continuity
2) It is possible for a function to be __________________ at and
not be _________________ at . Thus continuity does not imply
differentiability.
AP Calculus I
Notes 3.2
Basic Differentiation Rules and Rates of Change
The Constant Rule
Ex. 1:Use the definition of a derivative to find the derivative
of y = 7.
So, the Constant Rule is the derivative of a constant function
is ______
Find the derivative of: f(x) = 7 ______ s(t) = -3 ______ y =
kπ2_______
The Power Rule
Ex. 2: Use the definition of a derivative to find the derivative
of each of the following:
a) b) c)
So, the Power Rule is if n is a rational number, then the
derivative of a function is ___________
Ex. 3: Use the derivative rules to find the derivative of each
of the following:
a) b) c)
Ex. 4: Find an equation of the tangent line to the graph of when
.
**What two things do we need to write the equation of a line?
___________ and __________
The Constant Multiple Rule
Ex. 5: Use the definition of a derivative to find the derivative
of
So, the Constant Multiple Rule is if is a differentiable
function and c is a real number, the derivative of a function is
_____________
Ex. 6: Find the derivative of each of the following.
a) b) c) y = -
d) y = e) f)
The Sum and Difference Rules
The derivative of the sum (or difference) of two differentiable
functions is differentiable and is the sum (or difference) of their
derivatives.
Ex. 7: Find the derivative of each of the following:
a) b) c)
3.2 – Derivatives ( Day 2 )
Derivatives of Sine and Cosine Functions
Proofs: Graphically:
Graph of Graph of
Graph of Graph of
Derivative Rules of Sine and Cosine Functions
Ex. 8:
Find the derivative of the following:
a) y = 2 sin xb) y = 2x – cos xc) y = x2 + cos x
Evaluate the following:
d) e)
Derivative of the Natural Exponential Function
Ex. 9: Find the derivative of the following functions:
a) b)
Ex. 10: Determine the points on the curve at which the slope is
horizontal or vertical.
a)
b)
Ex. 11: Find the average rate of change over the interval from
of the function and then find the instantaneous rate of change at
each of the endpoints.
Derivative Worksheet
Find the derivatives of the following functions
1)2)
3)4)
5)6)
7)8)
9)10)
11)12)
13)14)
15)16)
17)18)
19)20)
AP Calculus I
Notes 3.3
The Product/Quotient Rules and Higher-Order Derivatives
Need for More Rules
1)
Given the function , find the derivative of by taking the
derivative of both terms, then simplify by multiplying the two
answers.
2)
Next, go back to and start by multiplying the terms and then
differentiate.
3)
Is it true that ?
The Product Rule:
Ex. 1: Find the derivative of each of the following:
a) b)
c) d)
Need for MORE Rules
1)
Given the function , find the derivative taking the derivative
of both functions, then simplify by dividing the two answers.
2) Next, simplify the expression, then differentiate.
3)
Is it true that ?
The Quotient Rule:
Ex. 2: Find each derivative:
a) b) c)
Ex. 3: Assume that and are differentiable functions about which
we know information about a few discrete data points. The
information we know is summarized in the table below:
-2
4
-1
5
6
-1
3
-5
1
7
0
-6
-3
8
-5
1
1
6
2
3
2
-1
5
1
?
Use your differentiation rules to determine each of the
following.
a)
If find b) If find
c) If find d) If , find
Derivatives of Trigonometric Functions/Proofs Using Quotient
Rule
Ex. 4:Find the derivative of each function.
a) b)
Higher Order Derivatives
The second derivative is the derivative of the first
derivative.
Ex. 5: Find the first, second, third, and fourth derivatives of
the function
Worksheet Derivatives 2: Finding derivatives and tangent
lines
Find the derivative of each of the following:
1)
2)
3)4)
5)6)
7)8)
9)Find the equation of the tangent line to at .
Rates of Change
Since derivative determines slope, it also determines the rate
of change of one variable with respect to another. There are many
applications of this concept: population growth rates, production
rates, water flow rates, velocity, acceleration, etc.
Recall that The average velocity is
Ex. 1:If a billiard ball is dropped from a height of 100 feet,
its height s at time t is given by the position function where s is
measured in feet and t is measured in seconds. Find the average
velocity over each of the following time intervals.
1.
2.
3.
Position / Velocity / Acceleration
You can obtain a velocity function by differentiating a position
function.
You can obtain an acceleration function by then differentiating
a velocity function.
Suppose that in the last example we wanted to find the
instantaneous velocity of the object when .
The velocity of the object at time t is the derivative of the
position function.
The position of a free-falling object (neglecting air
resistance) under the influence of gravity can be represented by
the equation
is the initial height of the object, is the initial velocity of
the object,
g is the acceleration due to gravity
Ex. 2: At time t = 0, a diver jumps from a diving board that is
32 feet above the water. The position of the diver is given by ,
where s is measured in feet and t is measured in seconds.
a. When does the diver hit the water?
b. What is the diver’s velocity at impact?
c. What is the diver’s acceleration at impact?
Ex. 3: A silver dollar is dropped from the top of the Empire
State Building, which is tall. Given the position function is .
(feet) or (meters)
a. Determine the position and velocity functions for the
coin
b.
Determine the average velocity on the interval .
c.
Find the instantaneous velocity when and .
d. Find the time required for the coin to hit the ground.
e. Find the velocity of the dollar just before it hits the
ground.
Ex. 4:A projectile is shot upward from the surface of the earth
with an initial velocity of 120 meters per second. What is its
velocity after 5 seconds? 10 seconds? What is it’s acceleration at
the same time periods?
39
Position/Velocity/Acceleration - Worksheet
·
Position equation, s, is given by the equation .
· Velocity is the derivative of position ( Speed is absolute
value of velocity )
· Acceleration is the derivative of Velocity
Practice Problems:
1. Bugs Bunny has been captured by Yosemite Sam and forced to
walk the plank. Instead of waiting for Yosemite Sam to finish
cutting the board from underneath him, Bugs finally decides just to
jump. Bugs’ jumps up at 16 ft/sec from a plank 320 feet high.
a. What is the position equation and velocity equation?
b. When will Bugs hit the ground?
c. What is Bugs’ velocity at impact?
d. What is Bugs’ speed at impact?
e. When does Bugs reach a maximum height?
f. What is Bugs’ maximum height?
g. Find Bugs’ velocity at 1 second.
h. Find Bugs’ average velocity from the time he jumped until the
time he landed.
i. What is Bug’s acceleration at 1 second.
2. Spider-Man goes to the top of a building 320 feet high to
prevent Dr. Octopus from destroying the Earth. He knocks Dr.
Octopus unconscious and throws him up in the air at 128 ft/sec.
a. Find the position, velocity, and acceleration as functions of
time.
b. How long will it take for Dr. Octopus to reach his maximum
height?
c. What is the maximum height that Dr. Octopus reaches?
d. When does Dr. Octopus hit the ground?
e. With what velocity does Dr. Octopus hit the ground?
f. What was his acceleration at the time of impact?
3. A student throws his Calculus book straight up from the
ground at 490 m/sec.
a. Find the position, velocity, and acceleration as functions of
time.
b. How long will it take the book to reach its maximum
height?
c. What is the maximum height the book goes?
d. When does the book hit the ground?
e. With what velocity does the book hit the ground?
f. With what acceleration does the book hit the ground?
4. A student drops his graphing calculator from the top level of
the parking garage 64 feet high.
a. Find the position, velocity, and acceleration as functions of
time.
b. When does the calculator hit the ground?
c. With what velocity does the calculator hit the ground?
d. What was the calculator’s average velocity from the time it
was dropped until it hit the ground?
e. What was the acceleration of the book half way down
Particle Motion
In discussing motion, there are three closely related concepts
that you need to keep straight. There are:
Ex. 1: If represents the position of a particle along the x-axis
at any time t , then this means…
1) “Initially” means when __________=0.
2) “At the origin” means _______________=0.
3) “At rest” means _______________=0.
4) If the velocity of the particle is positive, then the
particle is moving to the ____________.
5) If the velocity of the particle is ________________, then the
particle is moving to the left.
6) To find the average velocity over a time interval, divide the
change in ______________ by the change in time.
7) Instantaneous velocity is the velocity at a single moment
(instant!) in time.
8) If the acceleration of the particle is positive, then the
_________________ is increasing.
9) If the acceleration of the particle is ________________, then
the velocity is decreasing.
10) In order for a particle to change direction, the
_________________ must change signs.
Particle Motion
Velocity GraphRepresentation of the particle’s movement along
the x axis.
6
7
5
7
6
5
5
4
4
4
2
1
3
3
1
1
2
1. The particle is moving left and speeding up since velocity is
negative and the acceleration is also negative.
2. The particle is moving left at a constant speed ( at that
instant ) since velocity is negative and acceleration is 0.
3. The particle is moving left and slowing down since velocity
is negative and acceleration is positive.
4. The particle is at rest since velocity is 0 even though
acceleration is positive.
5. The particle is moving right and speeding up since velocity
is positive and acceleration is also positive.
6. The particle is moving right at a constant speed since
velocity is positive and acceleration is 0.
7. The particle is moving right and slowing down since velocity
is positive and acceleration is negative.
Ex. 2: The data in the table below give selected values for the
velocity, in meters/minute, of a particle moving along the x-axis.
The velocity v is a differentiable function of time t.
Time t (min)
0
2
5
6
8
12
Velocity (meters/min)
-3
2
3
5
7
5
1)
At , is the particle moving to the right or to the left? Explain
your answer.
2)
Is there a time during minutes when the particle is at rest?
Explain your answer.
3)
Use data from the table to find an approximation for and explain
the meaning of in terms of the motion of the particle. Show the
computation that lead to your answer and indicate units of
measure.
4)
Let denote the acceleration of the particle at time . Is there
guaranteed to be a time in the interval such that ? Justify your
answer.
Ex. 3: The graph below represents the velocity , in feet per
second, of a particle moving along the x-axis over the time
interval seconds.
1)
At seconds, is the particle moving to the right or the left?
Explain your answer.
2) Over what time interval is the particle moving to the left?
Explain your answer.
3)
At seconds, is the acceleration of the particle positive or
negative? Explain your answer.
4)
What is the average acceleration of the particle over the
interval ? Show the computations that lead to your answer and
indicate units or measure.
5)
Is there guaranteed to be a time t from such that ? Justify your
answer.
Ex. 4: A particle moves along the x-axis so that at time t
seconds its position in meters is given by:
1)
At , is the particle moving to the right or to the left? Explain
your answer.
2)
At , is the velocity of the particle increasing or decreasing?
Explain your answer.
3)
Find all the values of for which the particle is moving to the
left.
4) Find the velocity of the particle when the acceleration is
0.
Ex. 5: A particle moves along the x-axis such that its position
in feet can be modeled by the equation , where represents the time
in seconds, .
a) Find the velocity and acceleration function.
b) Find the initial position, velocity and acceleration of the
particle.
c) What is the velocity of the particle when the acceleration is
zero?
d) When is the particle moving to the right? Justify your
answer.
HW – Particle Motion 1 and 2 (Calculator may be used)
1) The position of a particle (in inches) moving along
the x-axis after t seconds have elapsed is given by
the following equation:
s = f(t) = t4 – 2t3 – 6t2 + 9t
a) Calculate the velocity of the particle at time t.
b) Compute the particle's velocity at t = 1, 2, and 4
seconds.
c) When is the particle at rest?
d) When is the particle moving in the forward (positive)
direction?
e) Calculate the acceleration of the particle after 4
seconds.
2.)The motion of a grizzly bear stalking its prey, walking left
and right of a fixed point in feet per second, can be modeled by
the motion of a particle moving left and right along the x-axis,
with a position equation of:
s (t) = sin t - t2
a) Identify the velocity equation that represents the bear's
motion.
b) Determine how fast the bear was traveling at t = 7
seconds.
c) In what direction is the bear traveling at t = 5
seconds?
d) How fast is the bear accelerating at a time of 2 seconds?
AP Calculus I
Notes 3.4
The Chain Rule
Exploration
1)
Use the basic power rule to find given . Next, multiply out and
then differentiate. Do you notice any similarities or
differences?
2)
Now, use the basic power rule to find given . Next, multiply out
and then differentiate. Do you notice any similarities or
differences?
3)
Next, use the basic power rule to find given . Next, multiply
out and then differentiate. Do you notice any similarities or
differences?
4)
Next, use the basic power rule to find given . Next, multiply
out and then differentiate. Do you notice any similarities or
differences?
When differentiating functions with inside parts and outside
parts, we must use The Chain Rule.
The Chain Rule
If is a differentiable function of u and is a differentiable
function of x, then is a differentiable function of x and
or
Ex. 1:Use the Chain Rule to find the derivative:
a) b) c)
d) e)
The General Power Rule
If , where u is a differentiable function of x and n is a
rational number, then
Ex. 2:Find all points on the graph of for which and those for
which does not exist. Then graph to verify your answers.
Ex. 3: Find the derivative of each of the following:
a) b)
Ex. 4: Find the second derivative of the following:
a) b)
Ex. 5: Find the derivative of each of the following:
a) y = tan 3xb) f ( x ) = cos ( x2 – 1 )
c) d)
f) e)
Theorem – Derivative of the Natural Logarithmic Function
Let be a differentiable function of .
1)2)
Ex. 6: Differentiate each of the following logarithmic
functions
a)b)
c)d)
Ex. 7: Use logarithmic properties to differentiate the following
functions.
a)b)
Because the natural logarithm is undefined for negative numbers,
you will often encounter expressions of the form . When you
differentiate functions in the form , do everything as usual.
Ex. 8: Find the equation of the tangent line for at .
Theorem – Derivatives for Bases Other than e
Let a be a positive real number () and let u be a differentiable
function of x.
1) 2)
3) 4)
Ex. 9: Find the derivative of each of the following:
a) b) c)
AP Calculus I
Notes 3.5
Implicit Differentiation
So far, most functions have been expressed in explicit form.
Some, however, are defined implicitly.
For example:
Implicit FormExplicit FormDerivative
It is not always possible, however, to solve for y explicitly.
For example, . In these cases, we must use implicit
differentiation.
The key to finding implicitly is understanding that the
differentiation is happening with respect to x.
· When you differentiate terms involving x alone, you can
differentiate as usual.
· When you differentiate terms involving y, you must apply the
Chain Rule.
Ex. 1: Differentiate each of the following:
a) b)
c) d)
Guidelines for Implicit Differentiation
1. Differentiate both sides of the equation with respect to
x.
2.
Collect all terms involving on the left side of the equation and
move all other terms to the right side of the equation.
3.
Factor out of the left side of the equation.
4.
Solve for by dividing both sides of the equation by the
left-hand factor that does not contain .
Ex. 2: Find given that .
Ex. 3: Find all points of horizontal and vertical tangencies for
the graph of .
Ex. 4: Determine the slope of the tangent line to the graph of
at the point (,).
Ex. 5: Determine the slope of the normal line of at the point
.
Ex. 6: Find :
Ex. 7: Given , find .
Ex. 8: Given , find .
AP Calculus I
Notes 3.6
Inverse Trigonometric Functions and Differentiation
Ex. 1: Evaluate each of the following:
a) b) c)
d) e) f)
Ex. 2: Use right triangles to evaluate the following
expressions:
a) Given , find b) Given , find
Derivatives of Inverse Trigonometric Functions
Proof:
Let be a differentiable function of .
Ex. 3: Differentiate each of the following:
a) b) c)
Ex. 4: Differentiate
Ex. 5: Write the equation of the tangent line to at .
AP Calculus
Notes 3.6
Inverse Functions
A function can be written as a set of ordered pairs:
FunctionInverse Function
Definition of Inverse Function
A function is the inverse of the function if:
for each in the domain of and for each in the domain of
Ex. 1: Show that the functions are inverses of each other
and
Ex. 2: Find the inverse of
Derivative of an Inverse Function Investigation:
1. Graph the functions (for ) and .
2. Verify that the two functions are inverses:
3. Write the ordered pair for at and determine the inverse point
on .
4. Using the feature on the calculator to determine the slope at
the ordered pairs found in the step above. Is there any
relationship between the two slopes?
5. Repeat the step above for the ordered pairs a) and then b) .
Is there any relationship between and , which is the same as ?
a)
b)
6. Make a conjecture about the derivative of inverse
functions:
Derivative of an Inverse Function:
Let be the inverse of , a differentiable function:.
Ex. 3: Let a)
What is the value of when
b)
What is the value of when
Ex. 4:
AP Calculus I
Notes 3.7
Related Rates
Another important use of The Chain Rule is to find the rates of
change of two or more related variables that are changing with
respect to time.
For example, when a balloon is inflated, there are numerous
characteristics (variables) of the balloon that are changing as
time goes on. Here are a few:
Ex. 1: Suppose x and y are both differentiable functions of t
and are related by the equation . Find when x= 1, given that = 2
when x = 1.
Guidelines for Solving Related Rate Problems
1. Identify all given quantities and quantities to be
determined. Make a sketch and label.
2. Write an equation involving the variables whose rates are
given or are to be determined.
3. Using the Chain Rule, implicitly differentiate both sides of
the equation with respect to time t.
4. After completing Step 3, substitute into the resulting
equation all known values for the variables and their rates of
change. Then solve for the required rate of change.
Ex. 2: A pebble is dropped into a calm pond, causing ripples in
the form of concentric circles. The radius r of the outer ripple is
increasing at a constant rate of 1 foot per second. When the radius
is 4 feet, at what rate is the total area A of the disturbed water
changing?
Ex. 3:Air is being pumped into a spherical balloon at a rate of
4.5 cubic inches per minute. Find the rate of change of the radius
when the radius is 2 inches.
Ex. 4:The radius, r, of a circle is increasing at a rate of 2cm
per minute. At the instant the radius of the circle is 6cm,
find:
a) The rate of change of the area.
b)The rate of change of the circumference.
Ex. 5:A television camera at ground level is filming the
lift-off of a space shuttle that is rising according to the
position equation , where s is in feet and t is in seconds. The
camera is 2000 feet from the launch. Find the rate of change in the
angle of elevation of the camera 10 seconds after lift-off.
Ex. 6:Liquid is dripping into a conical cup at the rate of 2.5
cubic inches per minute. The cup has a height that is always twice
the radius. How fast is the liquid level rising in the cup when the
liquid is 2 inches deep?
CALCULUS AB Name ________________________
Related Rate Problems
Solve the following related rates problems. Show all work and
circle your answer.
1. Air is leaking out of a spherical balloon at the rate of 3
cubic inches per minute. When the radius is 5 inches, how fast is
the radius decreasing?
2. A cone-shaped paper cup is being filled with water at the
rate of 3 cubic centimeters per second. The height of the cup is 10
cm and the radius of the base is 5 cm. How fast is the water level
rising when the level is 4 cm?
3. A 25-foot ladder leans against a vertical wall. If the bottom
of the ladder is slipping away from the base of the wall at the
rate of 1 foot per second, how fast is the top of the ladder moving
down the wall when the bottom of the ladder is 7 feet from the
base?
4.
A cylindrical tank of radius 10 feet is being filled with wheat
at the rate of 314 cubic feet per minute. How fast is the depth of
the wheat increasing? (Think about what is for a cylinder)
5. Oil from an uncapped oil well in the ocean is radiating
outward in the form of a circular film on the surface of the water.
If the radius of the circle is increasing at the rate of 2 meters
per minute, how fast is the area of the oil film growing when the
radius reaches 100 meters?
6. A conical tank is dripping oil into a cylindrical tank. The
cone has a diameter of 6cm and a height of 7cm.
a. Find the rate in which oil is leaving the conical tank when
the height is 2.4 cm and is decreasing at a rate of 0.18
cm/min.
b. Find the rate in which the height of the cylindrical tank is
increasing given the tank has a radius of 5cm and a height of 4 cm.
(Think about how the radius is changing over time in a
cylinder)
Calculus - Chapter 3 Review
Evaluate the following:
1.2.
Use the derivative rules to find the derivative of each of the
following functions.
3.4.
5.6.
7.An ice cube is melting at a constant rate such that it’s sides
are changing at . Find the rate in which the volume is decreasing
at the instant the side length is .
8.Find all the points on the graph of where there is a
horizontal and a vertical tangent line.
Find each derivative.
9.10.Find
11.12.Find :
13.Find the equation of the tangent line to at .
CALCULATOR PAGE - Use the chart and the functions below to find
the derivatives.
14.15.
Find the intervals of differentiability:
16.17.18.
19. The temperature T of food put in a freezer is where t is in
hours. Find the rate of change of the temperature after 3
hours.
20. If and , then what is ?
For questions 21 – 23: A particle is moving along the x-axis
where, for all values of time t for , the position can be modeled
by the function . Determine the following:
21.Find the velocity and acceleration at time .
22.What is the average velocity of the particle between the two
times in which the acceleration is zero?
23.What is the position when the velocity is first zero?
24. Sand is being poured onto the ground in the shape of a cone
whose height is always twice the radius.
When , what is the rate of change of the height if sand is
poured at a rate of 16 cubic feet per minute? The volume of a cone
is .
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