Week 3

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MATH 104 Small Class #3 Week of September 27, 2021 Questions marked with ( ) are more involved than the other questions. Questions with multiple stars are harder and should only be attempted after the rest was completed. 1 Rates of change and definitions. 1. A flight company data suggests that the total dollar cost of a certain flight is approximately C ( x ) = 0 . 005 x 3 - 0 . 38 x 2 + 120 x dollars, where x is the number of passengers. Below we show the graph and a table with values. 50 100 150 200 250 5 , 000 10 , 000 15 , 000 x y x 150 151 152 155 160 C ( x ) 11137.5 11177.1 11216.4 11332.4 11520 x 170 180 190 200 225 250 C ( x ) 11874.5 12204 12511.5 12800 13457.8 14062.5 a. (Calculator allowed) The corresponding flight currently has 150 passengers. If we add 100 more passengers, what is the average extra cost per passenger ? This corresponds to the average rate of change of C ( x ) between x = 150 and x = 250. What about adding 50 extra passengers instead? How about just 10? Definition(Average rate of change). The average rate of change of f ( x ) be- tween x = a and x = b = a + h is f ( b ) - f ( a ) b - a = f ( a + h ) - f ( a ) h . If x represents time and f ( x ) position on an axis, then this is the average velocity. b. Same question with adding just one extra passenger. c. Compute the instantaneous rate of change of C ( x ) at x = 150 defined by MC (150) = lim h ! 0 C (150 + h ) - C (150) h . Compare it to the result of b .
MC is called the marginal cost , and is used as an approximation of the extra cost generated when increasing the demand by 1. Definition(Instantaneous Rate of Change). The instantaneous rate of change of f ( x ) at x = a is the limit of the average rate of change between x = a and x = b as b approaches a , i.e. lim b ! a f ( b ) - f ( a ) b - a = lim h ! 0 f ( a + h ) - f ( a ) h . If x represents time and f ( x ) position on an axis, then this is the instantaenous velocity. Definition (Di erentiable, derivative). A function f ( x ) is di erentiable at x = a if lim x ! a f ( x ) - f ( a ) x - a exists . If it is the case, then we write f 0 ( a ) = lim x ! a f ( x ) - f ( a ) x - a exists, it is called the derivative of f ( x ) at x = a . A function is di erentiable if it is di erentiable at every point in its domain. We sometimes write d f d x instead of f 0 to make it clear that the variable is x . For the sake of computations, it is useful to replace x with t + a in the limit, and we get lim x ! a f ( x ) - f ( a ) x - a = lim t ! 0 f ( t + a ) - f ( a ) t . Definition(Tangent line). If the function f ( x ) is di erentiable at x = a then it admits a tangent line going through the point ( a, f ( a )). The slope of this line is f 0 ( a ), therefore we use the point-slope form to write the equation of the tangent line as: y = f 0 ( a )( x - a ) + f ( a ) . 2. Let f ( x ) = p 3 x + 1 and let a = 8. a. Use the definition of the derivative to find f 0 ( a ). b. Determine the equation of the line tangent to the graph of f at the point ( a, f ( a )). 3. Last week, we have shown that f ( x ) = | x | is continuous everywhere. Show that it is not di erentiable at x = 0. We have f is di erentiable at x = a ) f is continuous at x = a ) f has a limit at x = a. These implications only go in one direction!
4. The previous question shows a function that is continuous but not di erentiable. Find a function that has a limit at x = 0 but is not continuous at x = 0? 5. A love story, part 3. Skiing season started! As a family, R, L, and their kid K went on a ski trip. Unfortunately, the air is extremely foggy, so much so that they cannot see their own feet or the mountain. In spite of this, at the end of the day, the whole family remembers the shape of the slopes they skied down. How is that possible? The only information they have to go o of is the angle of their feet as they skiied down the mountain. By just being conscious of the angle of one's feet with the horizontal, one can detect bumps, steep slopes, plateaus, etc... Mathematically, skis ! tangent line , angle of skis ! slope of the tangent line ! derivative . 1 2 3 4 5 1 2 x y a. Assuming that the mountain is the graph of a function y = f ( x ) we know that f 0 ( x ) is the slope of the skis. Match the following: I. f is increasing 1. f 0 ( x ) < 0 II. f is decreasing 2. f 0 ( x ) = 0 III. f is at a maximum 3. f 0 ( x ) > 0 IV. f is at a minimum b. Sketch the derivative of the function above. Moral of the story. The angle of the tangent line (aka the derivative) is enough to know exactly the shape of a graph. The only thing that the family does not know, it their absolute altitude. Mathematically, the derivative of a function tells us everything about the function, up to a constant ( f ( x ) and f ( x ) + c have the same shape, and will yield the same derivatives). 6. ( ) Use the graph of f in the figure to do the following. a. Find the values of x in (0 , 3) at which f is not continuous.
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