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Find the derivative of: a. F(x) = ln( √4+x²) x b. Y = ln ∣-1 + sin x∣ ∣ 2 + sin x∣ Find an equation of the tangent line to the graph at the given point: x + y -1 = ln (x²+ y²) (1,0) Use logarithmic differentiation to find dy/dx: Y = √x²- 1 x²+1 Find the following: ∫ x4+x-4 x²+2 dx Solve the differential equation: Dy/dx = 2x (0,4) x²-9 Solve: e² ∫ e 1 dx x ln x Find the average value of the function over the given interval: F(x) = sec πx 6 [0,2] A population of bacteria is changing at a rate of dP/dt = 3000 1 + .25t Where t is the time in days. The initial population when t = 0 is 1000. Write an equation that gives the population at any time t, and find the population when t =3 The formula C = 5/9(F-32) where F≥-459.6 represents Celsius temperature, C as a function of Fahrenheit temperature, F. a) b) c) d) Find the inverse function of C. What does the inverse represent What is the domain of the inverse If C= 22° then what does F = Find the following: Y = ln ( 1+ ex) ( 1- ex) Find the equation of the tangent line to the graph at the given point: Y = xex – ex (1,0) The value, V, of an item t years after it is purchased is V= 15,000e-.6286t 0≤t≤10. Find the rate of change of V with respect to t when t=1 and t=5 Solve: √2 ∫ xe-(x²/2) dx 0 A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process. The flow rate R (in liters per hour) at time t (in hours) is given in the table: T 0 1 2 3 4 R 425 240 118 71 36 Ln R a) Use regression capabilities of a calculator to find a linear model for the points (t, ln R). Write the resulting equation in the fom ln R=at + b in exponential form. a) Use a definite integral to approximate the # of liters of chemical released during the 4 hours. Find the derivatives of: a) f(t) = t3/2log2√t + 1 b) h(x) = log3 x√x – 1 2 Integrate: ∫ 32x dx 1 + 32x e ∫(6x – 2x) dx 1 In a group project in learning theory, a mathematical model for the proportion P of correct responses after n trials was found to be: P = .86 1+e-.25n a) Find the limiting proportion of correct responses as n approaches ∞ b) Find the rates at which P is changing after n = 3 and n= 10 trials Find the derivative : Y = ½[ x √4-x² + 4 arcsin (x/2)] Y = 25arcsin (x/5) - x√25 - x² An airplane flies at an altitude of 5 miles toward a point directly over an observer. Consider Θ and x as shown in the figure: a) Write Θ as a function of x. b) The speed of the plane is 400 mph. Find dΘ/dt when x = 10 miles and x = 3 miles 1/√2 ∫arccos x dx 0 √1-x²