Iowa State University



Related Rates (3.10)Step 0: Draw a picture of what’s happeningStep 1: Write down changing functions of timeStep 2: Related by? Write down formula that relates the changing functions of timeStep 3: THE QUESTION -> translate to math (be specific)Step 4: The best thing to do with these problems is to develop a pattern and stick to it. Follow this set of steps or a set you developed on your own. This will help you a lot on the final and on this exam.1)A hot air balloon rising straight up from a level field is tracked by a range finder 500 ft. from the liftoff point At the moment the range finder’s elevation angle is π/4 the angle is increasing at the rate of .14 rad/min. How fast is the balloon rising at that moment?2) When a circular plate of metal is heated in an oven, its radius increases at the rate of .01 cm/min. At what rate is the plate’s area increasing when the radius is 50 cm?3) A 13-ft ladder is leaning against a house when its base starts to slide awa. By the time the base is 12 ft from the house the base is moving at the rate of 5 ft/sec.a. How fast is the ladder sliding down the wall then?b. At what rate is the area of the triangle changing?c. At what rate is the angle θ between the ladder and the ground changing then?Linearization (3.11)Find a linearization at a suitably chosen integer near x0 at which the given function and its derivative are easy to evaluate. Then calculate x0.fx= x2+2xx0=.1fx= 2x2+4x-3x0= -.9fx= 3xx0=8.5CHALLENGE PROBLEMShow that the linearization of fx=1+xkat x=0 is Lx=1+kxExtreme Values of Functions (4.1)Find the value of the absolute and local extrema on the given interval and where they occur. If no interval is given, the domain is all defined reals. DON’T DO THE FIRST DERIVATIVE TEST HERE. Please 1)fx=4-x2-3 ≤ x ≤ 12)Fx=-1x2.5 ≤ x ≤ 23)fx= -sinx-π2≤x≤5π64) y=x-4x5) y=xlnxThe First Derivative Test (4.3)Answer the following questions about the functions in the following exercises:On what intervals is f increasing or decreasing?Identify the function’s local extreme values in the given domain and say where they occurWhich of the extreme values, if any, are absolute?Sketch a graph based on the information above.fx=2x-x2-∞<x≤2gx=xlnx 0≤x≤10gx=x-2x2-1 -1≤x<1fx=-25-x2 -5≤ x≤5 hx=extanx -π/2≤x≤π/2 (extra)Concavity and Curve Sketching (4.4)Identify the inflection points and local maxima and minima of the following graph. Identify the intervals on which the function is concave up and concave down.Use the steps above to graph the following functions without using your calculators (don’t actually graph it, but follow those steps as if you were).y=x4-2x2y=sinxcosx0≤x≤πL’Hopital’s Rule (4.5)Indeterminate Forms:00 ∞∞(∞*0)(∞-∞)001∞∞0Once you don’t get an indeterminate form, you stop!limt→0sin5t2tlimx→∞(ln2x-ln?(x+1)) limx→1+x1x-1 limx→0ex+x1x limx→0+xx Applied Optimization (4.6)This is very similar to related rates! The more you develop a system of how to solve these problems, the easier it will be for you in the future.You have been asked to design a one-liter can shaped like a right circular cylinder. What dimensions will use the least material?A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions? (7)Antiderivatives (4.8)Find all solutions for the antiderivative F(x) given its derivative f(x): fx=-32x-52 fx=-sec3x22 fx= e-x5 25secxtanxdx (1+tanx2)dx 1x-5x2+1dx (1/x- 5/(x^2+1)dxFind the solution to the initial value problems:dydx=2x-7 y2=0dydx=1x2+xy2=1dsdt=1+costs0=4CHALLENGE: y’’=2t3y’1=1y(0)=0Answer Key:Related Rates: dydt=140ft/mindAdt=π a. dydt=-12ft/secb. dAdt=-59.5ft/sec2dθdt=-1rad/secLinearization: Lx=2xL.1=.2 Lx= -x+2L.9=1.1Lx=x12+43L8.5=4924fsdExtreme Values of Functions: Absolute max: (0,4)Absolute min: (-3,-5) Absolute max: (2,-.25)Absolute min: (.5,-4) Absolute max: (π2,1)Absolute min: (-π2,-1) Local max: (0,0)Absolute min: (4,-4) Absolute min: (1e,-1e)The First Derivative Test: 2x-x^2Increasing: (-∞,1] Decreasing: [1,2]Local max: (1,1) local min: (2,0)Absolute max: (1,1)–x^2-6x-9Increasing: [-4,-3]Decreasing: [-3,∞)Local max: (-3,0)local min: (-4,-1)Absolute max: (-3,0)(x-2)/(x^2-1)0<x<1Increasing: [0,1)Local max at x=0 Absolute min: 2-3,343-6Sqrt(25-x^2)Increasing: [-5,0]Decreasing: [0,5]Local max: (0,5)Local min: (-5,0) (5,0)Absolute max ““Absolute min ““Concavity and Curve Sketching: Increasing: (-∞,-1) U (1,∞)Decreasing: (-1,1)Local max: (-1, 27/7)Local min: (1, -27/7)CCU: (0,∞)CCD: (-∞,0)Inflection: (0,0)Local min: x=-1 and x=1Local max: (0,0)Inflection: x= 1/3x=-1/3Local min: x=0Absolute min: x=3π/4Local max: x=π Absolute max: x= π4 CCU: (π/2,π)CCD: (0,π)L’Hopital’s Rules:5/2ln2e1Applied Optimization: r=5.42 h=10.84 200m X 400m ................
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