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Chapter 3

Differential Calculus

3.1 Newton’s contribution to Calculus

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3.2 Introduction to Differential Calculus through two

examples

Newton’s remarkable contribution to Calculus was in fact motivated by two particular prob-lems related to his well-known work on the law of motion of objects.

• Calculating the rate of change of position (velocity) or of velocity (acceleration) ofmoving objects.

• Calculating the tangent to curves, and in particular trajectories of objects

Let’s now explore the two problems Newton considered, through two simpler examples.

3.2.1 Instantaneous velocity: cats falling

Cats (nearly) always land on their feet. However, whether they survive or not depends onhow fast they are falling at the time they hit the ground. We will now calculate the velocityof a cat falling from the roof of a small house (4m high).

The height of the cat, as a function of time, is given by the formula

h(t) = 4 − 4.9t2 if h(t) > 0

h(t) = 0 otherwise

The following table gives values of h at regular time intervals.

61

62 CHAPTER 3. DIFFERENTIAL CALCULUS

t (sec) h(t) (meters)0 4

0.1 3.9510.2 3.8040.3 3.5590.4 3.2160.5 2.7750.6 2.2360.7 1.5590.8 0.8640.9 0.0311.0 01.1 01.2 0

To calculate the velocity of the cat at each point in time, we can approximate it as thequantity:

Let’s now complete the table, and draw the resulting function v(t)

t (sec) v(t)0

0.1 0.000000000000000000000.20.30.40.50.60.70.80.91.01.1

Notes:

•

•

3.2. INTRODUCTION TO DIFFERENTIAL CALCULUS THROUGH TWO EXAMPLES63

Question 1: We want to find a better approximation to the velocity of the cat at a particularpoint, for example when t = 0.5 seconds. How would we do this?

Definition: The instantaneous velocity of an object is given by the formula

Question 2: Looking at the graph of v(t), it appears that v is a linear function of t. Canwe prove this mathematically?


3.2.2 Secants and Tangents

See Section Guided Work 5

3.3 Formal definition of derivatives

Textbook pages 165-170

Instantaneous velocities, rates of change and slopes of curves are examples of derivatives.

3.3.1 Definitions

Definition:

• The derivative of a function f(x) at a particular point x = c is given by the quantity.... and denoted as f ′(c).

• The derivative of f(x) at the point c only exists if f(x) is continuous at the point c

• One can define a new function f ′(x) which associates at each point x the derivative off at the point x.

• The function f ′(x) is called the derivative of f .

Examples: The derivative of the function f(x) = x2 at the point x = 1:

3.3. FORMAL DEFINITION OF DERIVATIVES 65

Examples: The derivative of the function f(x) = x2:

Examples: The derivative of the function f(x) = 2x + 1 at the point x = 2.5:

Examples: The derivative of the function f(x) = 2x + 1:

3.3.2 Geometrical interpretation of derivatives

Slope:

Example: f(x) = 2x + 1.


Example: f(x) = x2.

Important consequences (1):

Example:

Important consequences (2):

Example:

3.3. FORMAL DEFINITION OF DERIVATIVES 67

Check your understanding of Lecture 8

• Instantaneous velocity:Find the instantaneous velocity of the cat at t = 0.1 seconds by successive approxima-tions, as we did at t = 0.5 seconds, and verify that you recover the value predicted bythe formula for v(t) at t = 0.1.

• Velocity when the cat hits the ground:

– By solving the equation h(t) = 0, find the time at which the cat hits the ground.

– What is the velocity of the cat at that time?

• Generalization of the problemWe want to repeat the whole calculation by considering cats falling from an arbitraryheight h0 (instead of setting h0 = 4). In that case, h(t) = h0 − 4.9t2.

– First, find the time at which the cat hits the ground.

– Also, find the velocity of the cat as a function of time using the formula for v(t)involving limits.

– Finally, calculate the velocity of the cat as it hits the ground as a function of itsinitial height.

– Suppose cats have little chance of survival if they hit the ground at more than 20meters/second. What’s the highest they can fall from without dying?

• Acceleration:The acceleration of the cat is the rate of change of velocity, and is given by the formula

a(t) = lim∆t→0

v(t + ∆t) − v(t)

∆t

Using the formula for v(t) derived in the lectures, find the function a(t). What is specialabout a(t)?

• Graphical interpretation of derivatives Using the fact that the derivative f ′(x) is,at each point x, the slope of the function f(x), try to guess what the derivatives of thefollowing functions look like

– f(x) = x3

– f(x) = ex

– f(x) = cos(x)

– f(x) = ln(x)


3.3 How to calculate derivatives

Generally, we will not use the formal definition of derivatives to calculate f ′(x) for everyfunction f . Instead, we will only calculate the derivative of the standard functions this way,and then use derivative laws to calculate the derivatives of more complex functions.

3.3.1 Derivatives of constant functions

Rule: Let f(x) = K be a constant function. Then f ′(x) = 0.

Proof:

Graphical interpretation:

3.3.2 Derivatives of linear functions

Rule: Let f(x) = ax + b. Then f ′(x) = a.

Proof:

Graphical interpretation:

3.3. HOW TO CALCULATE DERIVATIVES 69

3.3.3 Derivative laws

Rule:

Examples:

• f(x) = x2 + x + 1

• f(x) = 5x2 + 2

Proof of (1)

Note:

3.3.4 Geometrical Application: Finding the maximum or minimum

of a parabola

A while back, we used the method of “completing the square” to determine the position ofthe maximum or minimum of a parabola. Now, we will use derivatives to do it and recover


the same result.

Idea: Given a function f(x) = ax2 + bx + c, where is the maximum or minimum of thegraph?

Example: Find the maximum or the minimum of the function f(x) = 4x2 − 2x + 1.


3.3.5 Physical Application: Deriving the law of motion of falling

objects

Newton’s Law of gravitation applied on Earth can be rewritten as

On Earth the constant g = 9.81m/s2. What is the height as a function of time of an objectfalling from a height h0 with zero initial velocity?

3.3.6 Derivatives of power law functions


Rule:

Examples:

• f(x) = x6:


• f(x) = 2x3

• f(x) =√

x

• f(x) = 1x

• f(x) = − 4x3

Proof: Proving this formula will be done later in the lecture.

3.3.7 Derivatives of polynomial functions

Rule:

Examples:

• f(x) = 4x3 − 2x

• f(x) = 10x8 + 6x6 − 2x4 + 3x2 − 1

• f(x) = 2 + x + x3 − 2x10 + x20

Proof:


3.3.8 Application: optimizing the shape of a swimming pool

You are managing a public aquarium (for instance, the one in Monterey), and want toconstruct a large new one that hold 10m3 liters of water, with a depth of 1m. Since glasswalls are expensive, you want to minimize the surface area of the walls, for the volume ofwater and depth given above. What is the optimal shape of the aquarium to construct?(Note that the aquarium is open-top, and that the bottom is cemented).



• Formal definition of derivatives:Using the formula (a + b)3 = a3 + 3a2b + 3ab2 + b3, expand (x + h)3. Then, using theformal definition of derivatives, find the derivative of f(x) = x3.

• Derivatives of power functionsWhat is the derivative of the function f(x)?

– f(x) = 3x5

– f(x) = 2x−1/2

– f(x) = 5x3

• Derivatives of constant, linear and polynomial functionsWhat are the derivatives of the following functions?

– f(x) = π + 1

– f(x) = x + 2

– f(x) = 3x2 − 1

– f(x) = 2x2 − x + 1

– f(x) = x3 + 3x − 2

– f(x) = −4x4 + 2x + 1

– f(x) = 15x5 + 1

4x4 + 13x3 + 1

2x2 + x + 1

– f(x) = a0 + a1x + a2x2 + a3x

3 + a4x4

• Maximum/minimum of parabola.Using derivatives, find the minimum/maximum of the following functions:

– f(x) = 3x2 + 4x − 1

– f(x) = 2x + x2 − 3

75

3.3.7 Derivatives of basic trigonometric functions


Rule:

•

•

Proof:

Graphical interpretation


3.3.8 Derivative of the natural exponential function


Rule:

Proof:

3.4 Derivative rules

3.4.1 Product rule


The derivative of the product of two functions is not equal to the product of the deriva-tives. Instead, one must use The Product Rule.

Rule:

Examples:

• f(x) = (x + 1)(3x + 2)

3.4. DERIVATIVE RULES 77

• f(x) = (x3 + 2x + 3)(4x4 + 3x2 − 1)

• f(x) = 3x6(2x3 + 1)

• f(x) = sin(x) cos(x)

• f(x) = ex cos(x)

Proof


3.4.2 Application: Graphing polynomials with derivatives

Now that we know how to calculate derivatives of complicated polynomials, and products offunctions, we can begin to combine our knowledge of derivatives and of signs tables to knoweven more about the functions before sketching them.

Example: Let’s study the function f(x) = (x + 1)(x2 − 3)

• First, do a standard signs table for f(x)

• Then calculate f ′(x)

• Factor f ′(x) and find the points at which f ′(x) = 0.

• Do a signs table for f ′(x), and conclude on whether f(x) is an increasing or decreasingfunction of x.

• Graph f(x) corresponding to your study, making sure to indicate all of the importantpoints in the graph.

3.4. DERIVATIVE RULES 79


• Derivatives of trigonometric functions:Using the formal definition of derivatives, show that (cos(x))′ = − sin(x). Hint: youwill need to use cos(a + b) = cos(a) cos(b) − sin(a) sin(b).

• Product Rule (1):Using the product rule, solve Textbook problems page 192-193 numbers 1,3,5,15,25,29,3

• Product Rule (2):Use the product rule to find the derivatives of

– f(x) = ex sin(x)

– f(x) = [cos(x)]2

– f(x) = e2x (Hint: write e2x = (ex)2)

• Graphing with derivativesStudy the functions f(x) = (x2 − 1)(x + 1) and g(x) = 3x(x + 1)(x − 2) using themethod described in the lecture, making sure in each case to draw both signs tablescorrectly, and finish the problem by sketching the functions.


3.4.3 The Reciprocal Rule

Rule:

Examples:

• f(x) = 1x+2 :

• f(x) = 1x3+2x :

• f(x) = sec(x) = 1cos(x) :

Proof:

3.4 DERIVATIVE RULES 81

3.4.4 The Quotient Rule


The quotient rule is used to calculate derivatives of functions which are defined as the ratioof two other functions.

Rule:

Examples:

• f(x) = x+1x−2

• f(x) =√

xx2+1

• f(x) = ex

x

• f(x) = tan(x)


• f(x) = cot(x)

Proof:

3.4.5 Application: maximizing your garden space

You want to build a small garden next to your house, with a wooden fence around and somelawn. The cost of the fence is $50 / linear foot; you only need it for 3 sides of the garden.The cost of the lawn is $5 / square foot. The cost of the work is a total of $500. You have$4000 to spend. What shape should your garden be to maximize the lawn area, and how bigwill it be?


GardenHouse

W

L


(continued)



• Reciprocal ruleWhat is the derivative of the function f(x)?

– f(x) = 13x+1

– f(x) = 12x2−4

– f(x) = 1ex

(Note: you will need to use a formula from the lecture notes)

• Quotient ruleWhat are the derivatives of the following functions?

– f(x) = x+1x3+3x−2

– f(x) = ex+1ex−1

– f(x) = x2−1x+1

– f(x) = tan(x)/ sin(x)

• Graphing with derivativesStudy the following function, using all that we have learnt so far:

f(x) =x2 − 4

x + 1

In particular,

– Calculate the limit of the function at +∞, −∞ and 0.

– Draw a signs table and identify the roots of the function (the points at whichf(x) = 0) and the asymptotes.

– Calculate the derivative of the function

– Draw a signs table for the derivative, and identify the intervals in which f(x)increases and those in which f(x) decreases.

– Deduce an accurate sketch of the function f(x).


3.4.6 Application: Determination of the optimal clutch size for a

given species


Mathematical modeling has proved a great success towards a better understanding of an-imal behavior. One example of the kind of studies typically done is presented here. Thefollowing model is an attempt to understand from a mathematical point of view what deter-mines the optimal number of offsprings per litter for various species to guarantee the highestreproduction rate.

Let’s consider two parents and N offsprings. We measure the resources (food, water, etc...)brought by the parents to the whole clutch as the number R (R is a fairly loosely determinedquantity. It could be for instance, the number of fish brought back by seals to their pups,or the mass of meat brought back by lions to their cubs, etc..). The available resources peroffspring is therefore the number x = R/N . Clearly, if x is too small, the probability ofan offspring surviving is very low, and the parents would have been better off having lessoffsprings. If x is too big, the probability of each offspring surviving is very high, but doesn’tkeep on increasing much with x (there is a saturation effect); in this case, the parents couldhave afforded having more offsprings without compromising their survival probability toomuch. So we can expect that there will be an optimal value of x, i.e. an optimal clutch sizegiven the available resources R, to maximize the number of surviving offsprings. We will nowtry to model this idea mathematically.

The number of surviving offsprings S(x) can be written as the number of born offspringstimes their survival probability p(x):

The survival probability p(x) is usually modeled as a sigmoid function (i.e. an S-shapedfunction).


Let’s now consider the function p(x) = x2

x2+k2

The first step is to check that it is indeed a Sigmoid function.

This means that the number of surviving offsprings is

and is a function with a graph that looks like this:

So what is the value of x that maximizes the number of surviving offsprings s(x) for a givenamount of resources brought by the parents?


This implies that the total number of offsprings the parents should have is

Note:

• The available resources R can change as a result of several factors:

• The value of k, which is the turnover point in the sigmoid function depends on thefragility of the offsprings: this varies a lot across species.

• What are the differences between this model and a model in which offsprings are left tofend off for themselves, when parents do not raise them (for instance, fish, or spiders,or turtles)

3.4.7 The Chain Rule


The Chain Rule is used to calculate derivatives of composed functions.

Rule:


Examples applied to polynomials.

• Find the derivative of u(x) = (3x2 − 1)2.

• Find the derivative of u(x) = (2x2 − 3x)7.

General rule for powers of functions:


Examples applied to radicals and other power laws

• Find the derivative of u(x) = (x6 − 1)1/3

• Find the derivative of u(x) =√

3x3 + x

• Find the derivative of u(x) = 1f(x) for any function f(x)

• Find the derivative of u(x) =√

f(x) for any function f(x)

Examples applied to the exponential function

• Find the derivative of h(x) = eg(x) for any function g(x).


• Deduce the derivative of h(x) = eKx where K is constant

• Deduce the derivative of h(x) = ax where a is constant

• Deduce the derivative of h(x) = ex2+1.

Examples applied to sin and cos

• Find the derivative of h(x) = sin(3x)

• Find the derivative of h(x) = cos(Kx) where K is constant


• Find the derivative of h(x) = sin(sin(x))

Nested chain rules...

• Find the derivative of u(x) =(√

x2 + 1 + 1)2

• Find the derivative of h(x) =(

sin(√

x2 + 1))2


3.4.8 Derivative of the inverse of a function


In some situations, a function is defined as the inverse of another function:

Examples:

• ln (x) is defined as the inverse of the exponential so if y = ln(x) then x = ey

• arcsin is defined as the inverse of sin so if y = sinx then x = arcins(y)

• cos

• tan

When this is the case, to calculate the derivative those functions we use the inverse rule.

Rule:

Examples:

• Let f(x) = ex, what is the derivative of ln(x) = f−1(x) ?


• Let f(x) = arcsin(x), what is f ′(x)?

• What is the derivative of arctan(x)?

Proof:


3.4.9 Derivatives of logarithmic functions


A logarithmic function in base a, f(x) = loga(x), is defined as the inverse of the expo-nential function in base a. So, using the inverse rule we find that

Examples:

• The derivative of the natural logarithm is

• The derivative of log2(x) is

Chain rule for logarithms: Given a function f(x), what is the derivative of ln[f(x)]?

Examples:

• f(x) = ln(3x + 1):

• f(x) = ln(sin(x)):


• f(x) = ln(√

x2 + 1):

• f(x) = ln[(x2 + 3x + 1)(x4 − 1)3]:



• Chain rule: Any of the Textbook problems page 209 number 8-28; 40-46.

• Exponential functions: Any of the Textbook problems page 221 number 14-34

• Inverse Rule:Using the inverse rule, find the derivative of f(x) = arccos(x).

• Derivatives of logarithmic functionsTextbook problems page 234 number 23, 27, 31, 33, 39, 43, 49

• Miscellaneous derivatives problemsTextbook review problems page 243 number 1,3,5,7,15,23,30,36

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