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Find the derivative of: a. F(x) = ln( √4+x²) x b.Y = ln ∣-1 + sin x∣ ∣ 2 + sin x∣

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Presentation on theme: "Find the derivative of: a. F(x) = ln( √4+x²) x b.Y = ln ∣-1 + sin x∣ ∣ 2 + sin x∣"— Presentation transcript:

1 Find the derivative of: a. F(x) = ln( √4+x²) x b.Y = ln ∣-1 + sin x∣ ∣ 2 + sin x∣

2 Find an equation of the tangent line to the graph at the given point: x + y -1 = ln (x²+ y²) (1,0)

3 Use logarithmic differentiation to find dy/dx: Y = √x²- 1 x²+1

4 Find the following: ∫ x 4 +x-4 dx x²+2

5 Solve the differential equation: Dy/dx = 2x (0,4) x²-9

6 Solve: e² ∫ 1 dx e x ln x

7 Find the average value of the function over the given interval: F(x) = sec πx [0,2] 6

8 A population of bacteria is changing at a rate of dP/dt = t Where t is the time in days. The initial population when t = 0 is Write an equation that gives the population at any time t, and find the population when t =3

9 The formula C = 5/9(F-32) where F≥ represents Celsius temperature, C as a function of Fahrenheit temperature, F. a)Find the inverse function of C. b)What does the inverse represent c)What is the domain of the inverse d)If C= 22° then what does F =

10 Find the following: Y = ln ( 1+ e x ) ( 1- e x )

11 Find the equation of the tangent line to the graph at the given point: Y = xe x – e x (1,0)

12 The value, V, of an item t years after it is purchased is V= 15,000e t 0≤t≤10. Find the rate of change of V with respect to t when t=1 and t=5

13 Solve: √2 ∫ xe -(x²/2) dx 0

14 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: T01234 R 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.

15 Find the derivatives of: a)f(t) = t 3/2 log 2 √t + 1 b)h(x) = log 3 x√x – 1 2

16 Integrate: ∫ 3 2x dx x e ∫(6 x – 2 x ) dx 1

17 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

18 Find the derivative : Y = ½[ x √4-x² + 4 arcsin (x/2)] Y = 25arcsin (x/5) - x√25 - x²

19 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

20 1/√2 ∫arccos x dx 0 √1-x²


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