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