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32 Linear Approximation—Using the Tangent Line An application of tangent line at a point (a, f(a)) is to approximate the function values near this point. For example, if f(x) = x2, what is f(1.98), i.e., 1.982? Assuming we don’t square 1.98? directly, we can’t find the exact answer, but we can find a close-enough answer by using the fact that 22 = 4. This is how:Recall that an equation of the tangent line at (a, f(a)) is y – f(a) = f (a)(x – a), i.e.,y =and we transform this equation a little bit tof(x) y = f(x)y = f (a)(x – a) + f(a)af(a)(a, f(a))and we have what is called the linear approximation or tangent line approximation of f at a.Take our example, given f(x) = x2 approximate f(1.98) using a = 2.Solution: Here x is 1.98 and a is 2.More examples:Use linear approximation to estimate/(3.96)2 3. sin 1
33 Product Rule, Quotient Rule and Chain Rule Manipulation Previously, we have shown you how to use Product Rule, Quotient Rule and Chain Rule to find the derivative of a function, when applicable. Recall that:Product Rule:Quotient Rule:Chain Rule: orAs you can see, Product Rule is for derivative of a product of two functions, Quotient Rule is for derivative of a quotient of two functions and Chain Rule is for derivative of a composition of two functions.Example 1 Given f(x) = x2 and g(x) = 3x – 4, finda) (fg)(1) b) (f /g)(2) c) [f(g(x))]|x = 3 or d) (f + g)(4)What happen if you are only given some function values and some derivative function values (but not the functions themselves)? For example,Example 2 Use the table on the right to find the following:a) (fg)(1) b) (f /g)(2)c) [f(g(x))]|x = 3 or d) (f – g)(4)xf(x)g(x)f (x)g(x)12345678910
34 Differentiability at Cutoff Numbers of Piecewise-defined Functions Previously, we’ve shown how to show whether a piecewise-defined function is continuous at cutoff numbers or not. Now, we are going to show whether a piecewise-defined function is differentiable at cutoff numbers or not (first with the graph, and then without the graph of the function).Example 1 Given that f is continuous at x = –1, 1 and 3, determine whether f is differentiable at these numbers or not.Example 2 Given that f is continuous at x = –1, 1 and 3, determine whether f is differentiable at these numbers or not.
35 Position and VelocityIf an object travels along a one-dimension line (as in figures 10 where a car is driving eastward and returning westward and as in figure 11 where a ball is thrown directly upward and coming back down along the same line it goes up), we can find its average velocity and its instantaneous velocity provided that we know its position at any time, i.e., its position function, which is usually denoted by s(t).DefinitionFigure 10Figure 11Average velocity is defined as the difference of two positions divided by the difference of the two times (that yield the two positions). That is, mathematically,vavg =Instantaneous velocity is defined as the velocity of the object at an instant (i.e., at a certain time). Its mathematical definition of isDoes the definition for instantaneous velocity look anything familiar? (Hint: f (a) = )If you can spot it, the velocity v at t = t0 is the _______________, i.e., _____. Furthermore, even though t0 is some specific time, it’s totally an arbitrary time (it can be as well as t1, t2, t3, ect., i.e., at any time t), therefore, we actually have the relationship ________. In (other) words, __________________________________.Example: The position of a car is given by the equation s(t) = t3 – 6t2 + 9t,1. Find the average velocity between t = 1s and t = 3s? Between t = 2s and t = 4s?2. Find the velocity at time t.3. Find the instantaneous velocity at t = 0s, and at t = 1s.4. Find the velocity at t = 2s, and at t = 3s.
36 Position and Velocity (cont’d) Remember, if a car is moving eastward with a positive velocity, then going westward (i.e., the opposite direction), it will sustain a negative velocity. If it stops (even momentarily), its velocity will be _____. To find the total distance the car (or any object) travels, we must know when it’s going one direction, when it’s going the opposite direction and when it stops.1. Assume moving eastward sustains a positive velocity, when is the car is traveling eastward? When is the car moving westward? When the car stops (i.e., at rest)?2. Draw a diagram to represent the motion of the car?3. Find the total distance traveled by the car during the first 5 seconds.Of course, an object can travel in a two-dimensional (2D) plane or in the three-dimensional (3D) space which is more like the real world. We can find an object’s velocity provided that we know its position function. If the path traveled by the object is 2D/3D , the velocity is also 2D/3D, which is more difficult to find. We will not do them here but will explain the situation in brief:Velocity is really a vector. A vector is always depicted by an arrow when it’s drawn (see figure 12). Mathematically, a vector can be determined by (or written in terms of):1. Its magnitude (i.e., length) and its direction (counterclockwise from the positive x-axis), or2. Its horizontal component and its vertical component.Figure 12vy= 453vx= 3We can describe the vector, , as1. A vector with magnitude 5 and direction 53.2. A vector with horizontal component 3 and vertical component 4.When the path traveled by the object is 2D, the velocity at a given time is a tangential vector at that time (see figure 13), which is also 2D.A word that is closely related to velocity is ______. However, they are not the same. Velocity is a vector, which means it has a magnitude and a direction whereas speed is a scalar, which means it is a real number. So how are they related? Speed is simply the magnitude of velocity, i.e., the absolute value of velocity: speed = |velocity|.1. What is the speed of the car at t = 2s and t = 4s? |v(2)| = 3, |v(4)| = 9Figure 13