. The differential of y f (x)

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1 Calculus I - Prof D Yuen Exam Review version 11/14/01 Please report any typos Derivative Rules Of course you have to remember all your derivative rules Implicit Differentiation Differentiate both sides of the equation with respect to the input variable, making sure that d(other variable) the derivative involving any other variable would generate a Then solve d(input variable) for your desired derivative Typically the derivative result will have both the input and output variables in it To evaluate the derivative at a point, you will typically plug in both coordinates of the point To find a point with a particular slope, set the derivative to the desired slope and solve this equation simultaneously with the equation of the curve The slope of a normal line is the negative reciprocal of the slope of the tangent line Related Rates Carefully read through the word problem twice On the first reading, try to understand the general picture and draw a diagram On the second reading, transfer information onto your diagram Be sure to use consistent units of measurement and that your final answer has units Any given number is either a constant, a rate, or an evaluation number On the diagram, label the variables and the constants Anything that changes as you visualize your diagram in motion is a variable Do not put evaluation numbers on your diagram; place the evaluation numbers on a separate second evaluation diagram -- these will be used in the final step Identify the known rates and the desired rate Remember that the rate of change of d (a variable ) d something with respect to time is The speed of an object is dt dt of a variable that measures the distance of this object to a fixed point that this object is moving towards or away from This derivative is negative if the quantity is decreasing Getting the signs correct is especially important when there are many rates involved Write an equation relating the variables and constants The number of variables in your equation should be the number of known rates plus one Typical techniques are: Pythagorean theorem (esp when one side is a constant), similar triangles (such as shadows), and trigonometry (such as rotating lights and angles) d Differentiate both sides with respect to time ( dt everything) Remember to use the chain rule or implicit differentiation Finally, plug in the known rates and any given specific evaluation numbers for the variables Sometimes, you need to do more work to find the evaluation numbers of other variables based on given evaluation numbers of some variables; use the evaluation diagram Linearization and Differentials Rates The linearization of f ( at base point x a is L( f ( a) f '( a)( x a) The differential of y f ( is dy f '(

2 For error propagation problems, is the error in the measurement of x, and dy is the corresponding error in using y f ( to calculate y Relative error for a variable x is Extrema and Concavity A critical point of a function is a number in the domain where the derivative is 0 or DNE Local extrema at critical points or end points (but not every critical point is a local extremum) To find the absolute extrema of a continuous function on a closed finite interval, one method is to compare the function values at all the critical points and end points To find the absolute extrema of a function on any interval, find the critical points and use the first derivative test to understand the graph f is increasing when f ' 0 and f is decreasing when f ' 0 [First derivative test] A critical point is a local max when f ' changes from to, and is a local min when f ' changes from to f is concave up when f '' 0 and f is concave down when f '' 0 f has an inflection point where f '' changes sign When investigating increasing/decreasing and concavity using first derivative and second derivative line diagrams (sign charts), remember to include locations of vertical asymptotes as well as critical points and possible inflections, respectively Horizontal and Vertical Asymptotes f has a horizontal asymptote (HA) at the horizontal line y L if L (asymptote is to the right) or L (asymptote is to the left) x f has a vertical asymptote (VA) at the vertical line x a if or and/or or xa Candidates for vertical asymptotes usually come from where the denominator of the function is zero but the numerator is not zero Sketching a graph Use facts about domain, intercepts, horizontal asymptotes, vertical asymptotes, local extrema, increasing/decreasing, inflection points, concavity Put it all together and think It usually helps to draw the line diagrams for increasing/decreasing and for concavity synchronously above where you will sketch the graph It is usually better to pencil in any asymptotic parts of the graph first You may be required to sketch a graph given the function or given derivative information or other information Max-Min word problems Read the problem carefully, draw a diagram and assign variables Do not assign variables to constant quantities Set up an objective function (to be maximized or minimized), which might at first involve more than one input variable Also, write down any relations (constraints) among the input variables If the objective function is in more than one input variable, then you must use the constraints to eliminate down to one input variable xa x x

3 Note the domain of the objective function in this one input variable You usually find the domain by considering for what inputs does the function make sense This often involves looking at any constraints, if there are any Take a derivative and find the critical points in the domain Find the absolute maximum or minimum If the domain is a closed finite interval, then you could use the method of comparing the function values at the critical points and end points Or, in any case, you could use the first derivative to understand the graph (for example, you could be lucky and there is only one critical point and the signs of the derivative would tell you exactly where the desired absolute extremum is) Newton's Method f ( xn ) To numerically find a solution to f ( 0, the recursion is xn 1 xn f '( x ) n Practice Problems Your homework, worksheets, quizzes are also good sources dy 1 Find and the tangent line and normal line to: (a) xy x y 4 0 at ( 1,) (b) sin( x 4y) x y 0 at ( 0,0) Find a point where the curve y xy x 0 has a horizontal tangent A man is standing 5m from a standard intersection A car travelling at 10m/s passes through the intersection along the other road How fast is the distance between the man and the car increasing when the car is 1m past the intersection? 4 In # above, if the man rotates his head continuously to face the car, how fast is his head rotating? Be sure to give your answer with units 5 Car A is travelling towards an intersection at 5 ft/s and Car B is travelling away from the intersection on the other road at 0 ft/s What is the rate of change of the distance between the two cars when Car A is 100 ft from the intersection and Car B is 40 ft from the intersection? A lighthouse located 100 ft from a straight shoreline is rotating at revolutions per minute How fast is the spot of light on the shoreline moving when the spot of light is 00 ft from the lighthouse? 7 A ft woman is walking away from a 10 ft lamp post at night at ft/s How fast is the tip of her shadow when she is 7 ft away from the lamp post? 8 Find the linearization of: x 4x 5 at x (b) g( cos( at x 9 Find the differential dy : (a) y x 4x 5 (b) y xsin( 10 We measure a cube to have side length m with maximum possible error of 001m Use differentials to approximate the maximum possible error in calculating the volume 11 Find the absolute extrema of: x x 1 on [ 1, ] (b) g( cos( x on [ 0, ] 1 Find the critical points of: 1 x (b) g( x 1/ x x 0 7

4 1 Find all local extrema of: 5 4x x 18x 5 (b) g ( x 1/ ( x 1) 14 Find all inflection points of: 4 4 x x x x 1 (b) g( x x 15 Sketch a graph having the following information: f ( 4) 0, f '() 0, f ''() 0, 1,, vertical asymptote at x 0, x x f '( 0 for x 0 and for x ; f '( 0 for 0 x f ''( 0 for x 0 and for 0 x 4; f ''( 0 for x 4 1 Sketch a graph having the following information: f ( 0) 1, f ( ) 0, f '(1) 0, f ''() 0,,, vertical asymptote at x 0, x x f '( 0 for 0 x 1; f '( 0 for x 0 and for x 1 f ''( 0 for x ; f ''( 0 for x 0 and for 0 x 17 A farmer has 400 ft of fence and wishes to make 4 identical side by side pig rectangular pens in a 1 by 4 formation What dimensions will maximize the total area? 18 A farmer wishes to make 4 identical side by side rectangular pig pens in a 1 by 4 formation with each pig pen having area 10 square feet What dimensions will minimize the length of fence used? 19 A box with no top is to be feet wide If the volume is to be cubic feet, what should the other two dimensions be so as the minimize the material? 0 We want to make a box where the length is twice the width The bottom of the box costs 15 as much as the rest of the material If the box is to have volume 90 cubic feet, what dimensions will minimize the cost? 1 Write the recursion in Newton's method for numerically solving x x 1 0 Starting with x 1, find 0 x Leave it as a fraction Solutions to Practice Problems dy y 5 1 (a), tangent line is y ( x 1), normal line is y ( x 1) xy 1 5 dy cos( x 4y) (b), tangent line is y x, normal line is y x 4cos( x 4y) 1 dy y x Solve y x 0 along with y xy x 0 to get x The answer y x is there are two such points: (, ) and (, ) 10/1 m/s Use Pythagorean theorem 4 50/19 rad/s Use trigonometry (the tan function) 5 100/ 1100 ft/s Use Pythagorean theorem 400 ft/min Use trigonometry (the tan function) 7 15/ ft/s Use similar triangles 1 8 (a) L ( 5 8( x ) (b) L ( x ) (

5 9 (a) dy (x 4) (b) dy (sin( x cos( ) m [ V x, dv x ] 4 11 (a) Max is 1 at x 0, and Min is at x 1 (Critical points at x 0, ) (b) Max is at x and Min is at x (Critical points at x ) 0 0 7x 91x 1 (a) Critical points at x 0, 1, 1 (Derivative is 0 ( x 7) (b) Critical points at 1, 1 1 x 0, 1 1 (Derivative is / x ) 1 (a) Local max at x 1, Local min at x (b) Local min at x (a) Inflection Points at x, 1 (b) Inflection Points at x, ) 1 Other solutions possible, such as the vertical asymptote going to -infinity as x approaches 0 from the right

6 17 Each pen should be 5 ft by 40 ft (the sides that are "shared" have length 40 ft) The total area is 4000 square feet 18 Each pen should be 10 ft by 1 ft (the sides that are "shared" have length 80 ft) The total length of fence is 10 feet 19 Dimensions are ft wide by ft long by ft tall 0 Dimensions are ft wide by ft long by 5 ft tall xn xn 1 1 The recursion is x n1 xn x 1 /, x 1/ x 1 n

AB Calculus: Extreme Values of a Function

AB Calculus: Extreme Values of a Function Name: Extrema (plural for extremum) are the maximum and minimum values of a function. In the past, you have used your calculator to calculate the maximum and minimum