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Natural Exponent
Applications of Exponential Functions
Derivative of y=lnx
Derivative of y=e^x
Random
100
Solve the following equation (round to two decimal places): e^3x-2 + 5 = 8
e^3x-2+5 = 8 e^3x-2=3 lne^3x-2=ln3 3x-2=ln3 3x=ln3+2 x=(ln3+2)/3 x=1.03
100
There was an earthquake in toronto that measured 6.8 on the richter scale, and an earthquake in vancouver that measured 8.4 on the scale. How much more intense was the earthquake in vancouver?
= 10^1.6 =39.8 The earthquake in vancouver was 39.8 times greater than the one in toronto.
100
Differentiate. y=ln(5X^(2)e^(5X^(2)+2)
y=ln5 + lnX^2 + lne^(5X^(2)+2) =ln5 +2lnX + 5X^(2)+2 y'=2/X + 10X = (2+10X^(2))/X
100
Determine the derivative of the following: y=e^-2x^2 and y=(1-e^5x)^5
y'= -4xe^-2x^2 and y'=-25e^5x(1-e^5x)^4
100
What is the difference between a natural logarithm and a common logarithm? State their equations and in exponential form.
Natural Logarithm: base e y=lnX y=e^x Common Logarithm: base 10 y=logX X=10^y
200
Solve the following: a) ln(18t+5)=2 b) e^(2ln10) c) lne^(10^(-15))
a) e^2 = 18t+5 7.389=18t+5 t= 2.389/18 or t= 0.133 b) e^(ln10^(2)) =100 c)= 10^-15
200
A population of bacteria quadruples every 4 hours. Determine the amount of bacteria of a 15 g sample after 1 day. b) How long will it take for there to be a 2 million g sample?
A=15(4)^6 A=61440 =6.14 x 10^4 b) 2000000 = 15(4)^x/4 log(2000000/15)=log4^x/4 log(2000000/15)=(x/4)log4 log(2000000/15)/log4=x/4 x= 34 hours
200
Differentiate. y=ln(5X^(2)e^(5X^(2)+2)
y=ln5 + lnX^2 + lne^(5X^(2)+2) =ln5 +2lnX + 5X^(2)+2 y'=2/X + 10X = (2+10X^(2))/X
200
Differentiate. y=(1-e^5x)^5
y'=-25e^5x(1-e^5x)^4
200
Solve for X. a) ln(x+1)^2 = 2
x = 1.72
300
ln(e^(2)lne^(3)
ln3 +2
300
The population of Aliens from the planet tralfamadore doubles in size every two hours. 12 hours ago the population had 200 000 aliens. a) How many aliens are in the population now? b)When were there 600 aliens in the population?
a) there are 12 800 000 aliens now. b)about 28.76 hours ago
300
Find the equation of the tangent of the curve y=(2e^-3x)(ln(x+2)) at the point (-1,5).
40x-y+45=0
300
Differentiate. a) f(x) = e^x/(e^(x) - 1) b)f(x) = xe^(2x^(2)-1)+e^(-5x^(3)-8)
a) -e^x/(e^(x)-1)^2 b) e^(2x^(2)-1)+4x^(2)e^(2x^(2)-1)-15x^(2)e^(-5x^(3)-8)
300
Find the max and min values of on 0
Max value at x=1/4
400
Solve for x. e^(2x) - 8e^(x) + 5 = -10
e^x = 5 or e^x = 3 x=ln5 or ln3
400
A sink hole grows according to the equation D = Doe^(kt) where D is diameter (cm) t years after the start. After 3 years the diameter is 60.5 cm. After 8 years the diameter is 200.6 cm. Find the values of k and Do
k = 0.24 Do = 29.32
400
Find the equation of the tangent of the curve f(x)=ln[(1+x)(3+x^2)^4(2+3x^3)^3 at the point (1,0).
37x-4y-37=0
400
Find the maximum and minimum values of f(x)=2Xe^-5x+1
There is a max value at x=1/5 … see solution.
400
6.27=e^x
x=1.83
500
ln(8x)^1/2 +ln4x^2 -ln(16x)^1/2
Written on Board….ln(square root) 8 * x^2
500
y=(ln(x^(2)+4) - lnx)/2lnX
answer written on board.
500
Determine the equation of the tangent of the curve f(x)=e^(2x+1)-ln5 that is parallel to the line 4y-8x+9=0. (round decimals to two decimal places)
2x-y+0.7=0
500
Who discovered the constant e…. / what is it called?
Oiler!