' -Activit 1 MCV4U Un|t2 y Review of Prerequisite Skills for Unit # 2 (Derivatives) Working with the properties of exponents Simplifying radical expressions Finding the slopes of parallel and perpendicular lines Simplifying rational expressions. (add/subtract/multiply/divide and restrictions) Expanding and factoring algebraic expressions Evaluating Expressions Working with the difference quotient. O O C Homework: p. 62 # 1 I] The Derivative Function In the previous chapter we discussed the slope of a tangent line and rates of change using limits. Hm f(x+ h) f(x) haU h This limit has two interpretations: the slope of the tangent and the instantaneous rate of change. Limits are used so often in calculus they have be given a specific name and concise notation. The limit is called the Several accepted notations of derivatives are; N.B. f ' is itself a function and it equals the slope of the function f at the point ( x , f(x) ). Eg.l: Find the derivative of f(x) = x2 @ x = 4 . (using first principles) [ sol: -8 ] If we want to find the derivative of a function at numerous points it is tedious if we use the definition each time. So, we use the definition but keep the variable it instead of a specific number. MCV4U Eg.2: The derivative of a function at an arbitrary value. Find the derivative of f (x) = x2. (using first principles) [ f ' (x) = 2x ] Now solve forf ' (~11) , f ' (O) , f ' (5). NB. Notice that the derivative of a quadratic function (degree two) is a linear function (degree one). A) Determine the derivative with respect to x of each of the following functions: 0 r(x)=x ii) f(X)=x3 iii) f(x)=x"' iv) f(x)=x5 B) What pattern do you see developing. C) Predict the derivative of f(x) = x39 D) Generalize the derivative for f(x) = x" , when n is a positive integer? The Denition of the Derivative Function f(x+h)f(x) h The derivative of for) with respect to x is the function f ' (x) where f ' (x) = lim , provided that haO this limit exists. MCV4U We can now define (instantaneous velocity) as the derivative of the position function with respect to time. If the position function at time t is given by s(t), then the velocity at time t is given by; The process of finding the derivative of a function is called differentiation. A function is differentiable at "a" if f '(a) exists. u as n.b. If you can not draw a unique, defined tangent at "a" then the function is not differentiable at a . 3 ways derivatives fail to exist: Cusp Vertical Tangent Discontinuity The Normal of a Tangent: Homework: (p.73#1,5,6d,7b,9,10,12cd,15,19)