Chapter 3Differential Calculus3.1 Newton’s contribution to CalculusSee PowerPoint presentation or PDF3.2 Introduction to Differential Calculus thr ough twoexamplesNewton’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 veloc ity (acceleration) ofmoving objects.• Calculating the tangent to curves, a nd in particular trajectories of objectsLet’s now explor e the two problems Newton considered, through two simpler examples.3.2.1 Instant aneous velocity: cats fallingCats (nearly) always land on their feet. However, whether they survive or not depe nds onhow fast they ar e 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, a s a function of time, is given by the formulah(t) = 4 − 4.9t2if h(t) > 0h(t) = 0 otherwiseThe following table gives values of h at regular time intervals.6162 CHAPTER 3. DIFFERENTIAL CALCULUSt (sec) h(t) (meters)0 40.1 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 0To 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)00.1 0.000000000000000000000.20.30.40.50.60.70.80.91.01.1Notes:••3.2. INTRODUCTION TO DIFFERENTIAL CALCULUS THROUGH TWO EXAMPLES63Question 1: We want to find a better approximation to the velocity of the cat at a par ticula rpoint, for e xample when t = 0.5 seconds. How would we do this?Definition: The instantaneous velocity of an object is given by the formulaQuestion 2: Looking at the gr aph o f v(t), it appears that v is a linear function of t. Canwe prove this mathematically?64 CHAPTER 3. DIFFERENTIAL CALCULUS3.2.2 Secants and TangentsSee Section Guided Work 53.3 Formal definition of derivativesTextbook pages 165-170Instantaneous velocities, rates of change and slo pes of curves are examples of derivatives.3.3.1 DefinitionsDefinition:•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) = x2at the point x = 1:3.3. FORMAL DEFINITION OF DERIVATIVES 65Examples: The derivative of the function f(x) = x2:Examples: The derivative of the function f(x) = 2x + 1 a t the point x = 2.5:Examples: The derivative of the function f(x) = 2x + 1:3.3.2 Geometrical interpretation of derivativesSlope:Example: f(x) = 2x + 1.66 CHAPTER 3. DIFFERENTIAL CALCULUSExample: f(x) = x2.Important consequences (1):Example:Important consequences (2):Example:3.3. FORMAL DEFINITION OF DERIVATIVES 67Check 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 sec onds, 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 gro und.– 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 2 0meters/second. What’s the highest they ca n fall from without dying?• Acceleration:The acceleration of the cat is the rate of change of velocity, and is given by the formulaa(t) = lim∆t→0v(t + ∆t) − v(t)∆tUsing 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)68 CHAPTER 3. DIFFERENTIAL CALCULUS3.3 How to calculate derivativesGenerally, 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 mor e complex functions.3.3.1 Derivatives of constant functionsRule:Let f(x) = K be a constant function. Then f′(x) = 0.Proof:Graphical interpretation:3.3.2 Derivatives of linear functionsRule:Let f(x) = ax + b. Then f′(x) = a.Proof:Graphical interpretation:3.3. HOW TO CALCULATE DERIVATIVES 693.3.3 Derivative lawsRule:Examples:• f(x) = x2+ x + 1• f(x) = 5x2+ 2Proof of (1)Note:3.3.4 Geometrical Application: Finding the maximum or minimumof a parabolaA while back, we used the method of “completing the square” to deter mine the position ofthe maximum or minimum of a parabola. Now, we will use derivatives to do it and recover70 CHAPTER 3. DIFFERENTIAL CALCULUSthe same result.Idea: Given a function f(x) = ax2+ bx + c, where is the maximum or minimum of thegraph?Example: Find the maximum o r the minimum of the function f(x) = 4x2− 2x + 1.3.3. HOW TO CALCULATE DERIVATIVES 713.3.5 Physical A pplication: Deriving the law of motion of fallingobjectsNewton’s L aw of gravitation applied on Earth can be rewritten asOn Earth the constant g = 9.81m/s2. What is the height as a function of time of an objectfalling from a height h0with zero initial velocity?3.3.6 Derivatives of power law functionsTextbook pages 179-183Rule:Examples:• f(x) = x6:72 CHAPTER 3. DIFFERENTIAL CALCULUS• f(x) = 2x3• f(x) =√x• f(x) =1x• f(x) = −4x3Proof: Proving this formula will be done later in the lecture.3.3.7 Derivatives of polynomial functionsRule:Examples:• f(x) = 4x3− 2x• f(x) = 10x8+ 6x6− 2x4+ 3x2− 1• f(x) = 2 + x + x3−