Let f(3:)= $23ml5. Show that the average rate of change of f (with respect to 3:) on an arbitrary interval [a , b] is equal to the instantaneous rate of change of f at the midpoint of that interval. Show and explain your work. (Note: This will be true for any quadratic function, and not just for the given one.) The position of an object moving on a number line is given by s(t) =3 +3t. At which of the following instants is the object slowing down? Choose all that apply. Ot = -2 O t = - Ot=0 Ot = Ot = 2Consider the effect of gravity on falling objects. If you drop an object, from rest, on earth, it will travel d(t) =4.9t2 meters in t seconds. How high do you need to lift a ball (in meters), so that it will be traveling at 98 meters per second when it hits the ground?Assume the functions f and g are differentiable functions, with f(1) =6, f'(1) =-2, 9(1) =-1 and g' (1) = 3 . f(z) + 3ac Find an equation of the line perpendicular to the graph of F(ac) = at x=1 . 2- g(x)The curve y=ax + ba + c passes through the point (1, 2) . The line y=2ac +1 is tangent to the curve when x=0. What are the values of a , b and c? O a =1, b = 2, c=1 O a = 1, b = -2, c=1 O a = -1, b=2, c=-1 O a = -1, b = -2, c=1 O a =-1, b =2, c=1Which of the following lines is perpendicular to the tangent line of y: at the point (2 , 2) ? m3 O _:1:8 9' 5 Oy23mI4 _;r:4 3' 3 O _a':+4 3" 5 238 0y: In the following question, show all your steps, and do not use any differentiation rules from Section 3.3. 1 Let f($)=3:3 +5. :1: (a) Using the Denition of the Derivative from page 125, determine f ' (3:) . (b) Using the Alternative Formula for the Derivative from page 126, determine f'($) again (you should get the same answer as in (a) ). 126 Chapter 3 Derivatives which the limit exists, which means that the domain may be the same as or smaller than Secant slope is the domain of f. If J' exists at a particular x, we say that f is differentiable thas a deriva- tive) at x. If f exists at every point in the domain of f. we call / differentiable. If we write : = x + h. then & = c - x and h approaches O if and only if z approaches E. Therefore, an equivalent definition of the derivative is as follows (see Figure 3.4). This formula is sometimes more convenient to use when finding a derivative function, and focuses on the point : that approaches x. Alternative Formula for the Derivative Derivative of fat x is Calculating Derivatives from the Definition FIGURE 3.4 Two forms fur the diller- The process of calculating a derivative is called differentiation To emphasize the idea that differentiation is an operation performed on a function y = fux), we use the notation ence quotient. Derivative of the Reciprocal Function as another way to denote the derivative /'(x). Example I of Section 3. 1 illustrated the diff- ferentiation process for the function y = 1/x when x = a. For I representing any point in the domain, we get the formula & (4) =4 Here are two more examples in which we allow x to be any point in the domain of f- EXAMPLE 1 Differentiate () = = 1 Solution We use the definition of derivative, which requires us to calculate fix + A) and then subtract j(x) to obtain the numerator in the difference quotient. We have ( + h) Definition x+A-1 = lim = lim (x+h- 1)(x - 1) = lim -. -k Pok (xth - 1)(x - 1) Simplify. = 10 + h - 1( - 1 4 - IF Cancel & # 0 and evaluate. EXAMPLE 2 (a) Find the derivative of fur) = Vx for x 2 0. (b) Find the tangent line to the curve y = Vx ax = 4. 3.2 The Derivative as a Function 127 Derivative of the Square Root Solution Function (a) We use the alternative formula to calculate f's TVIN, 120 lim Me - Vi V - Vi = lim (V - V)( v + v) . 3 - V Caned and evaluate. (b) The slope of the curve at x = 4 is FIGURE 3.5 The curve y = Vix and its tangent line al (4, 2). The tangent line's slope is found by evaluating the derivative The tangent is the line through the point (4. 2) with slope 1/4 (Figure 3.5): al s = 4 (Example 2) y = 2 + 2(x - 4) y= + +1 Slope Notation Slapa =1 There are many ways to denote the derivative of a function y = f(r], where the indepen dent variable is x and the dependent variable is y. Some common alternative notations for he derivative include 15 The symbols d/ de and Do indicate the operation of differentiation. We read dy/ dx as "the derivative of y with respect to x." and alf/ de and (d/ di) ](x) as "the derivative of f with Slope respect to x." The "prime" notations y' and J' originate with Newton. The affix notations are similar to those used by Leibalz. The symbol dy/dy should not be regarded as a ratio; it is simply a notation that denotes a derivative. To indicate the value of a derivative at a specified number x = a, we use the notation For instance, in Example 2 -I- Vertical cordingis = 1 Graphing the Derivative FIGURE 3.6 We made the graph of We can often make an approximate plot of the derivative of y = f(x) by estimating the " = f'(x) in (b) by plotting slopes from slopes on the graph of f. That is, we plot the points (x, /(1)) in the ry-plane and connect the graph of y = fix) in tak The verti- them with a smooth curve, which represents y = fork call coordinate of B' is the slope at 8 and so un. The slope al E is approximately EXAMPLE 3 Graph the derivative of the function y = f(x) in Figure 3.fa. 8/4 - 1 In th) we see thil the male of change of f is negative for a between A" Solution We sketch the tangent lines to the graph of f at frequent intervals and use their and D', the rak of chun siting for a clopac to actimate the values of J'(x) at these points. We plot the corresponding (x, S'(2)) In the right of D'. pairs and connect them with a smooth curve as sketched in Figure 3.fit