1. Find the derivative of:
a. F(x) = ln( √4+x²) x
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:
∫ x4+x-4 dx
x²+2

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

6. Solve: e²
∫ 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 = 3000
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≥-459
The formula C = 5/9(F-32) where F≥ represents Celsius temperature, C as a function of Fahrenheit temperature, F.
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 =

10. Find the following:
Y = ln ( 1+ ex)
( 1- ex)

11. Find the equation of the tangent line to the graph at the given point:
Y = xex – ex (1,0)

12. The value, V, of an item t years after it is purchased is V= 15,000e-
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

13. Solve: √2
∫ xe-(x²/2) dx

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: T 1 2 3 4 R
425
240
118 71 36
Ln R
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.
Use a definite integral to approximate the # of liters of chemical released during the 4 hours.

15. Find the derivatives of:
f(t) = t3/2log2√t + 1
h(x) = log3 x√x – 1 2

16. Integrate:
∫ 32x dx
1 + 32x e
∫(6x – 2x) 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
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:
Write Θ as a function of x.
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