Calculus 1
Lesson- Definition of a Derivative
Name:_________________________________
Date:__________________________________
For the graph of a function?
For a circle:
secant
tangent
f(x) = x(x-2)(x-4)
A tangent
-touches but does not cross?
-touches at only one point?
How will we define a tangent line to the graph of y=f(x) at the point (c, f(c))?
Step 1:
Slope of the secant line through the point (c, f(c)) and a second point of the curve:
Step 2:
Slope of the tangent line at the point (c, f(c))
1
DEFINITION: If f is defined on an open interval containing c and if the limit exists, then the line passing
through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)).
Example: Find the slope of the tangent line to the graph of y = x² at the point (3, 9).
Note: If lim
x 0
f (c  h )  f (c )
  then the tangent line at (c, f(c)) is vertical.
h
Example: Find the slope of the tangent line to the graph y  3 x at x = 0.
y3 x
2
Test for Differentiability
If a function is not continuous at a certain point, then it is not differentiable at that point.
~Check to see if the function is continuous
f ( x )  f (c )
~If so, use the alternate derivative formula f ' ( x)  lim
to determine if the function has a left- and
x c
xc
right-handed limit that are the same.
~If so, the function is differentiable. If not, it’s not.
Test for Continuity
If a function is not differentiable at a certain point, then it is not continuous at that point.
Textbook definition: “f is differentiable on the closed interval [a,b] if
If f is not continuous at x=c then f cannot be differentiable at x=c.
But f can be continuous at c and still not be differentiable at x=c.”
Example: Find the derivative with respect to x of the function y = 2/x.
3
Calculus 1
Lesson- Derivative Shortcut Rules
Name:_________________________________
Date:__________________________________
Objective:
To learn the most common derivative rules
Constant Rule:
The derivative of a constant is 0.
Power Rule:
d n
x  nx n 1
dx
 
Examples:
f ( x)  x 5
f ( x) 
f ( x)  x10
Constant Multiple Rule:
1
x2
f ( x)  3 x
d
d
[cf ( x)]  c [ f ( x)]
dx
dx
Examples:
f ( x)  4 x 3
k ( x) 
f ( x) 
f ( x)  5 x
2
3x 2
m( x ) 
2
x
f ( x)  0.47 x100
5
 3x 
2
Sum and Difference Rules: The derivative of the sum (or difference) equals the sum (or difference) of the
derivatives.
d
[ f ( x)  g ( x)]
dx
f ( x)  x 3  4 x 2  9 x  46
2
g ( x)  5 x 2  5
x
Example:
Find
4
Derivative of the Sine/Cosine Functions:
d (sin x)
 cos x
dx
Special Trig Rules:
sin x
lim
x 0
x
Why do these work? Take a look at the graphs of each…
and
d (cos x)
  sin x
dx
1  cos x
0
x 0
x
lim
Proofs of Sine/Cosine Derivative Rules:
Examples:
g ( x)   cos x
h( x)  x  sin x
j ( x) 
2
 4 cos x
4
x
More Derivative Practice:
Check each using 2nd CALC 6 with an x-value of  .
1. Find the slope of the tangent to the graph of g (t )  2  3cos t at the point ( , 1)
2. Find the equation of the tangent line to the graph of y  3(5  x)2 at the point (5, 0).
3. Determine the points, if any, at which the function has a horizontal tangent line.
1
(a) y  x3  x
(b) y  2
x
5
Calculus 1
Lesson- Position Function and Velocity
Name:____________________________________
Date:_____________________________________
Objective:
To learn how to use derivatives as a means to solve projectile motion problems.
Velocity:
Derivative of the Position Function = Instantaneous Velocity Function
Position Function s(t) =
Example 1 A ball is dropped from a height of 100 feet. It’s height, in feet, t seconds later is given by
s(t) = -16t² + 100
What is the average velocity of the ball between 1 and 2 seconds after it is dropped?
change in dis tan ce
Note: Average velocity is
also Negative velocity means the object’s speed is in a
change in time
downward direction
What is the average velocity of the ball between 1 and 1.5 seconds after it is dropped?
What is the average velocity of the ball between 1 and 1.1 seconds after it is dropped?
What is the instantaneous velocity at exactly 1 second after it is dropped?
Note: Instantaneous velocity is found by finding the first derivative then subbing in a given value of t.
v(t) =
6
Example 2
A ball is thrown straight down from the top of a 220-foot building with an initial velocity of
-22ft/sec. What is its velocity after 3 seconds?
s(t)=
v(t)=
v(3)=.
What is its velocity after falling 108 feet?
Find the position (height above ground) after falling 108 feet.
Find the time t at which the ball is at this position.
Find the velocity at this particular t.
Example3
A ball is thrown straight down from the top of a 220-foot building with an initial velocity of
-22ft/sec. What is its velocity after 3 seconds?
7
Calculus 1
Lesson – Product and Quotient Rules
Name:____________________________________
Date:_____________________________________
The Product Rule
If f and g are two differentiable functions, then their product fg is also a differentiable function and
d
 f ( x) g ( x)  f ( x) g ( x)  g ( x) f ( x) .
dx
In words,
The derivative of a product is
the 1st function times the derivative of the 2nd plus the 2nd function times the derivative of the 1st.
Example 1
Differentiate: h( x)   6 x  5   x 3  3
Example 2
If g ( x)  x sin x , find g ( x) .
Example 3
y  sin x cos x
Find
dy
.
dx
8
The Quotient Rule
If f and g are two differentiable functions, then their quotient f/g is also a differentiable function (wherever
g ( x)  0 ) and
d  f ( x)  g ( x) f ( x)  f ( x) g ( x)
.

2
dx  g ( x) 
 g ( x)
In words,
The derivative of a quotient is
the denominator times the derivative of the numerator minus the numerator times the derivative of the
denominator, all “over’ the denominator –squared.
x2  2
, find f ( x ) .
2x  7
Example 4
If f ( x) 
Example 5
Differentiate t ( x) 
Example 6
y
sin x
.
x
dy
x3  3x  2
. Find
.
2
dx
x 1
9
dy
x3  3x  2
. Find
.
2
dx
x
Example 7
y
Example 8
Find f ( x ) if f ( x) 
Example 9
Differentiate
Example 10
5
.
4 x4
1 
2
y  x2  
.
 x x 1
 x2  x  3  2
f ( x)   2
  x  x  1
x

1


10
Derivatives of Trig Functions (Section 2.3)
If y  sin x,
dy
 cos x
dx
If y  cos x,
dy
  sin x
dx
If y  tan x,
dy

dx
If y  cot x,
dy

dx
If y  sec x,
dy

dx
If y  csc x,
dy

dx
Higher Order Derivatives
(Section 2.3)
The derivative of the first derivative of f is called the second derivative of f.
The derivative of the second derivative of f is called the third derivative of f.
And so on… for the nth derivative of f.
Notation:
Function
y
f(x)
First Derivative
y
dy
dx
Dx  y 
f ( x )
d
 f ( x)
dx
Second derivative
y 
d2y
dx 2
Dx 2  y 
f ( x)
d2
 f ( x) 
dx 2
Third derivative
y
d3y
dx 3
Dx3  y 
f   x 
d3
 f ( x) 
dx3
d4y
dx 4
Dx  y 
f
 x
d4
 f ( x) 
dx 4
Dx n  y 
f ( n)  x 
dn
 f ( x) 
dx n
Fourth derivative
y
(4)
.
.
.
n th derivative
4
(4)
.
.
.
y(n)
dny
dx n
11
Example 1
Find the second derivative of f ( x)  12 3 x .
Example 2
Find the second derivative of h(t )  4sin t  5cos t .
When our function is a position function , y = s(t), then
the first derivative is the
the second derivative is the
function = s’(t)
function =s”(t)
90t
where v is measured in feet per second.
4t  10
Find the vehicle’s velocity and acceleration at each of the following times:
Example 3
An auto’s velocity starting from rest is v(t ) 
(a)
1 second
(b)
5 seconds
(c)
10 seconds
12
Example 4
The graphs of f , f  and f  are shown on the same set of coordinate axes. Which is which?
Explain your reasoning.
13
Calculus 1
Lesson- Chain Rule
Name:____________________________________
Date:_____________________________________
Chain Rule
Let y be a differentiable function of u, y = f(u),
and u be a differentiable function of x, u = g(x).
d  f  g ( x)  d
d
Then,
  f (u )   g ( x) .
dx
du
dx
Think: (derivative of the outside) times (derivative of the inside)
dy dy du


In other symbols:
dx du dx
Find the first derivative of each:
y  sin  x 4  5 
1.
1
2.
y  sin 2 x  (sin x)2
5.
y  cos3 4t 2
8.
3.
y  x2  3
 
6.
f ( x)  x  3x  9 
g ( x)  3x  5 tan(2 x)7
9.
g ( x) 
4.
y
7.
y
10.
Find the derivative of the function f ( x) 
x
2
 2x
x

3
x 4
4
3
cos x
csc x
1
 2 
 cos x at the point  ,  .
x
2 
14
11.

Find an equation of the tangent line to the graph of f ( x)  9  x 2

2
3
at the point (1,4).
Graph the function and its tangent.
12.
Determine the points at which the graph of f ( x) 
x
has a horizontal tangent.
2x 1
15
Calculus 1
Lesson- Implicit Differentiation
Name:____________________________________
Date:_____________________________________
So far we have been working with functions in explicit form (equations solved for y in terms of x). Now we
will learn how to work with implicit forms of equations (equations not solved for y or not easily solved for y).
Explicit Form of y = function of x;
y written explicitly in terms of x
y
Implicit Form of y = function of x;
an equation that relates y to x but where y cannot
necessarily be isolated
1
x
xy  1
Implicit Differentiation
Step 1
Step 2
Step 3
Step 4
Note:
Differentiate both sides of the equation with respect to x.
dy
Collect all terms which contain
on one side of the equation and everything else on the other side.
dx
dy
Factor
out of all the terms on the one side.
dx
dy
Solve for
dx
dy
will be in terms of x and y.
dx
Example 1
Graph x 2  y 2  16 and find
dy
implicitly.
dx
Find all points where the graph has a horizontal tangent.
dy
Using implicit form of
:
dx
Using explicit form of
dy
:
dx
Using explicit form of
dy
:
dx
Find all points where the graph has a vertical tangent.
Using implicit form of
dy
:
dx
16
Example 2
a.
Find the first derivative of x 2 y  y 2 x  2 by implicit differentiation.
b.
Find f’(2)
Example 3
x2 y 2

1
6
8
a.
Graph the hyperbola:
b.
Find an equation of the tangent line to the graph of the hyperbola
Example 4
Find an equation of the tangent line to the astroid x
Example 5
Differentiate implicitly:
2sin x cos y  1
a.
b.
2
3
y
x2 y 2

 1 at the point (3,-2).
6
8
2
3
 5 at the point (8,1).
cot y  x  y
17
Calculus 1
Lesson- Related Rates
Name:____________________________________
Date:_____________________________________
Objectives:
Identify a mathematical relationship between quantities that are each changing.
Use one or more rates to determine another rate.
Process:
Step 1: Draw a diagram.
Step 2: Determine which quantities and rates are given, and which to be found.
Step 3: Identify the primary function to use. (Often this is a formula from geometry.)
Step 4: Differentiate with respect to the independent variable
Step 5: Write a related rates equation
Step 6: Substitute known quantities and solve for desired rate.
[NOTE: Do not substitute known quantities before this last step!]
Type 1: Explicit Function of One Variable
Examples
1.
Air is being blown into a sphere at the rate of 6 cubic inches per minute. How fast is the radius
changing when the radius of the sphere is 2 inches?
2.
The edge of a cube is increasing at a rate of 2 inches per minute. At the instant the edge is 3 inches, how
fast is the volume increasing?
18
3.
A point moves along the curve y   x  3 such that its x-coordinate is increasing at 4 units per second.
(a)
At the moment x = 1, how fast is the y-coordinate changing? Interpret your answer based on the
shape of the graph and the location of the point.
2
(b)
At the moment x = 1, how fast is the point’s distance from the origin changing?
Type 2 - Implicit Function of One Variable
4.
A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at
a rate of 1 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is
6 ft from the wall?
Type 3: Functions of Two Variables—2 rates given
5.
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?
Type 4: Functions of Two Variables—1 rate given--secondary equation needed
6.
A water tank has the shape of an inverted circular cone with base radius 2m and height 4 m. If water is
being pumped into the tank at a rate of 2m3/min, find the rate at which the water level is rising when the
water is 3 m deep.
19
Calculus 1
Unit 3- Lesson- Extrema and Critical Number
Name:____________________________________
Date:_____________________________________
Extreme Value Theorem
If f is continuous on a closed interval I = [a,b] then f has both a minimum and a maximum value on I.
f has a relative maximum at c or f(c) is a relative maximum of f if there is some open interval I containing c
on which f(c) is the maximum.
f has a relative minimum at c or f(c) is a relative minimum of f if there is some open interval I containing c on
which f(c) is the minimum.
Relative Extrema
Let I be any interval (closed or open) containing the x-value c.
The extreme values, or extrema, of a function f on I are defined by:
The absolute maximum of f on I is f(c) if f (c)  f ( x) for all x in I.
The absolute minimum of f on I is f(c) if f (c)  f ( x) for all x in I.
NOTE: A function need not have a maximum value or a minimum value on a given interval I.
Process
 Find the first derivative and plug the x-value of the coordinate of the relative min/max & solve for f’(x)
Or

Find the left- and right-hand side derivatives using lim
x c
f ( x )  f (c )
xc
Examples:
For each: determine any relative extrema and then determine the derivative at each relative extremum.
1.
f ( x)  x on the interval [-2,2].
2.
f ( x) 
1
on the interval [-2, 2].
x
3.
f ( x)  x 2 on the interval (-1, 2).
20
4.
f ( x)  x 3  3 x 2
5.
f ( x)  x
6.
f ( x)  sin x
7.
f ( x) 
2
3
8
x 4
2
Critical Values
A critical value is a number c in the domain of f for which f (c)  0 or f (c ) does not exist.
*Relative Extrema Occur Only at Critical Numbers
* One result of this theorem is that, on a closed interval, absolute extrema must occur at local extrema or at the
endpoints.
Long story short: Find the derivative, set = 0, solve for x, plug x back into original function.
Examples:
Find the critical value(s)
8.
f ( x)  x
10.
f ( x) 
2
3
8
x 4
2
9.
f ( x)  sin x
11.
y  2x 3
21
Method for Finding Absolute Extrema on a Closed Interval [a,b]
(1)
(2)
(3)
(4)
Find the critical numbers of f in (a,b).
Evaluate f at each critical number in (a,b).
Evaluate f(a) and f(b).
Compare: the least of these y-values is the minimum; the greatest is the maximum.
2x  5
on the interval [0, 5].
3
12.
Find the maximum and minimum values of f ( x) 
13.
Find the maximum and minimum values of f ( x)  x 2  2 x  4 on the interval [-2, 1].
14.
Find the maximum and minimum values of f ( x)  x3  12 x on the interval [0, 4].
15.
Find the maximum and minimum values of g ( x)  3 x on the interval [-1, 1].
16.
   
,
Find the maximum and minimum values of h( x)  sec x on the interval 
.
 6 3 
22
Calculus 1
Lesson- First Derivative Test
Name:____________________________________
Date:_____________________________________
Objective:
Learn about Increasing and Decreasing Functions & the First Derivative Test for Extrema
Graphically
f increasing
f decreasing
Numerically
When x’s go up, y’s go up.
When x’s go up, y’s go down
Sign of the Derivative
f ( x)  0
f ( x)  0
First Derivative Test
If f  changes from – to + at c, f has a relative minimum at (c, f(c)).
If f  changes from + to – at c, f has a relative maximum at (c, f(c)).
If f  doesn’t change sign at c, then f(c) is not a relative extremum for f.
Where “c” is a critical value.
Process:
Step 1 Find all critical numbers of f in the given interval; break the interval into smaller intervals using these
critical values, points of discontinuity, and the endpoints.
Step 2 Create a sign chart, by picking an x-value in each interval and finding the sign of f  there.
Step 3 For each sub-interval state whether f is increasing or decreasing there.
Example
1.
Tell where f ( x)  x3  2 is increasing, decreasing, and identify any relative extrema.
x3
is increasing, decreasing, and identify any relative extrema.
x2
2.
Tell where f ( x) 
3.
Use the graph of f  to tell where f is increasing, decreasing, and identify any relative extrema.
23
Calculus 1
Lesson- Concavity, Inflection, 2nd Derivative Test
Name:____________________________________
Date:_____________________________________
Objective:
To use derivatives to determine concavity and points of inflection
Concavity-
Function has a hill (concave down) or a valley (concave up)
Points of Inflection- Location at which a function goes from being concave down to concave up (or vice versa)
Concave Up
Concave Down
Inflection
Second Derivative Test for Concavity
Process:
1. Find the 1st derivative
2. Set = 0 and solve
3. Find the 2nd derivative
4. Set = 0 and solve (result could be a point of inflection)
5. Create sign chart
Guided Example
1 3 1 2
x  x
3
2
2
f ' ( x)  x  x
f " ( x)  2 x  1
0x x
0  2x  1
x  0,1
x  1 / 2
f ( x) 
2
Intervals
Test Values
Sign of f’(x)
Sign of f”(x)
Conclusion
 ,1
-2
+
Increasing
Concave down
(-1, -1/2)
-3/4
+
Increasing
Concave down
(-1/2, 0)
-1/4
+
+
Increasing
Concave up
0, 
2
+
+
Increasing
Concave up
Since the function went from concave down to concave up at x = -1/2, that must be the inflection point.
24
Examples
1.
Find the intervals where f ( x) 
6
is concave upward and concave downward.
x 3
2
Determine relative extrema, concavity, points of inflection.
2.
g ( x)  x 3  2 .
3.
  
h( x)  2 x  tan x ;   ,  .
 2 2
Let f be a function with f (c)  0 and the second derivative f  exists on an open interval containing c.
Then
if f”(c)>0, then f has a relative minimum at (c, f(c)),
if f”(c)<0, then f has a relative maximum at (c, f(c)),
if f”(c)=0, then test fails; must use the 1st Derivative Test for Extrema (could be max, min, or neither)
Find all the relative extrema
x
.
x 1
4.
f ( x)  x3  9 x 2  27 x.
5.
f ( x) 
6.
f ( x)  x 2  1.
7.
f ( x)  sin x  cos x.
25
Calculus 1
Lesson- Curve Sketching
Name:____________________________________
Date:_____________________________________
Objective:
To learn how to sketch a graph using calculus and without using a graphing calculator
Analyzing and Sketching the Graph of a Function
1. Decide domain and (where possible) range.
2. Determine x- and y-intercepts.
3. Note symmetry where applicable. (even function: f(-x) = f(x) -- symmetric across y-axis
odd function: f(-x) = -f(x) – symmetric across origin)
4. Find any points of discontinuity.
5. Find any vertical and horizontal asymptotes.
6. Find x-values where f and f are 0 or non-existent. Use these to get relative extrema and pts. of inflection.
7. Find end behaviors that are infinite.
Example 1 Polynomial Function
f(x) = x4 - 12x3 + 48x2 - 64x
Example 2
Rational Function
2
2( x  9)
f ( x)  2
 x  4
26
Calculus 1
Lesson- Optimization
Name:____________________________________
Date:_____________________________________
Optimization Problems—Applied Minimum and Maximum Problems
Method for Solving:
Step 1 Define variables needed to describe the quantity to be optimized.
Step 2 Write a primary equation for the quantity to be optimized.
Step 3 Use other info given in the problem to write a secondary equation that relates the variables used in the
primary equation. Solve this equation to get one variable in terms of the other. Use this equality to rewrite the primary equation in terms of one variable.
Step 4 Find a feasible domain for the function you wrote in Step 3, i.e., upper and lower bounds on the variable
that make sense for your problem.
Step 5 Find absolute or relative max or min of your function on the feasible domain.
Step 6 Write your answer in an English sentence.
Example 1
Sr. Karlien wants to create a rectangular garden. She has available 100 feet of fencing to make
the border. What are the dimensions of the plot with the largest area that can be enclosed with
this fence?
Example 2
Find 2 positive numbers whose product is 192 and whose sum is a minimum.
Example 3
A manufacturer wants to design an open box having a square base and a surface area of 108
square inches. What dimensions will produce a box of maximum volume?
Example 4
A box with a square base and open top must have a volume of 32,000 cubic cm.
Find the dimensions of the box that minimize the amount of material used
27
Calculus 1
Lesson- Rolle’s Theorem, MVT
Name:____________________________________
Date:_____________________________________
Objective:
To learn about and apply Rolle’s Theorem and the Mean Value Theorem
Rolle’s Theorem:
Given an interval (a, b) if f(a)=f(b) then there is a location in the interval where the first
derivative = 0.
MVT:
On the closed interval [a,b]:
f ' (c ) 
f (b)  f (a )
{used for finding average velocity)
ba
Determine if Rolle’s Theorem can be applied. If so, apply it. If not, explain why.
f ( x)  sin x [0,2 ]
1.
2.
f ( x)   x 2  3 x
[0,3]
Determine if Mean Value Theorem can be applied. If so, apply it. If not, explain why.
f ( x)  cos x  tan x
3.
4.
f ( x)  x 4  8 x
[0,2]
[0,  ]
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