Calculus & Vectors Vectors: Calculus: So let’s get started …

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Presentation transcript:

Calculus & Vectors Vectors: Calculus: So let’s get started … Unit 1: Introduction to Vectors Unit 2: Application of Vectors Unit 3: Lines & Planes Unit 4: Points, Lines & Planes Final Exam: June Calculus: Unit 1: Limits of Functions Unit 2: The Derivative Unit 3: Derivative Applications Unit 4: Curve Sketching Unit 5: Derivatives of Exponential & Trigonometric Functions Final Exam: April So let’s get started …

What is Calculus? A method for solving TWO simple geometric problems … The Area Problem The Tangent Problem How can we calculate the area under a curved line? How can we calculate the slope of a tangent line to a curve?

The Area Problem Any ideas? …. In the mid-17th Century, mathematicians such as Sir Isaac Newton and Gottfried Leibniz become focussed on finding the area under a curved line. Any ideas? ….

The Area Problem HINT: They used the sum of the area of rectangles to approximate the area. Which of these sums is closer to that of the area of the curve?

The Area Problem So …. as the width of the rectangles gets smaller and smaller (approaches zero), the sum of the areas gets closer and closer to that of the curve. The Area Problem is known as INTEGRAL CALCULUS and as fascinating as it is it is no longer part of the curriculum so unfortunately this is the last we’ll see of it. However … the process of “shrinking” a value to get closer and closer to an answer will be used A LOT!

The Tangent Problem Any ideas? …. Sir Isaac Newton and Gottfried Leibniz also wanted to know what the precise slope of the tangent to curve so that they could know the INSTANTANEOUS RATE OF CHANGE at a specific point on the curve. Any ideas? ….

The Tangent Problem Recall … You calculated the slope of a SECANT to a curve. Δ𝑦 Δ𝑥 𝑚 𝑃𝑄 = Tangent 𝑦 𝑄 − 𝑦 𝑃 𝑥 𝑄 − 𝑥 𝑃 = Secant You used this to calculate the AVERAGE RATE OF CHANGE of a function between to points.

The Tangent Problem Example: Application of Rates of Change The following graph shows the height of a ball, ℎ 𝑡 , in metres 𝑡 seconds after being dropped from a building.

The Tangent Problem Example: Application of Rates of Change We can calculate the AVERAGE SPEED of the ball between one and two seconds after being dropped by calculating the slope of the SECANT PQ.

The Tangent Problem Example: Application of Rates of Change The INSTANTANEOUS SPEED of the ball 1 second after being dropped is found by calculating the slope of the TANGENT at point P. Does anyone remember the method that you used to ESTIMATE the slope of the TANGENT?

The Tangent Problem Example: Application of Rates of Change You “shrunk” the distance to P. Example: Application of Rates of Change You picked spots that got closer and closer to P and calculated the slope of the secant formed by P and those spots. This is A LOT of work!!!

The Tangent Problem Investigating the slope of the secant PQ as Q → P (i.e. the horizontal distance, ℎ, between P & Q “shrinks to zero”) to estimate the slope of the of the tangent is known as DIFFERENTIAL CALCULUS and it what we will be focussing on this semester.

The Tangent Problem If we define P & Q using function notation then we get: 𝑚 tangent = lim ℎ→0 𝑓 𝑎+ℎ −𝑓(𝑎) 𝑎+ℎ−𝑎 = lim ℎ→0 𝑓 𝑎+ℎ −𝑓(𝑎) ℎ This is called the DIFFERENCE QUOTIENT.

So what do we have to review before we get started? Calculating the slope and the equation of a line. (Grade 9) Function notation. (Grade 11) Domain and Range. (Grade 11) Calculate the slope of a secant → average rate of change. (Grade 12) Estimate the slope of a tangent → instantaneous rate of change. (Grade 12) Simplify rational expressions by factoring. (Grade 11) Simplify rational expressions with RADICALS. (NEW)

Before we begin … Please come grab a textbook from the front of the class, find your name on the list and write the number found on the first page CLEARLY beside your name. NOTE: You are responsible for returning your ASSIGNED textbook at the end of the semester. The replacement value of a textbook is $75. Grab a FORMULA SHEET from the front of the class.

PRACTICE QUESTIONS: p.2-3 #1abf, 2adef, 3, 4ac, 5-9, 10b, 11a, 12a As you just finished Advanced Functions, I’m assuming most of this is fresh in your brains, so I will leave you to it. Please put any questions you have on the board and we will take these up as a class. QUESTIONS: p.2-3 #1abf, 2adef, 3, 4ac, 5-9, 10b, 11a, 12a