Does y = e- - 2 exist? What is the domain & range of the function if it does exist?
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![Example 1
Does y = e9-x* - 2 exist? What is the domain & range of the function if it does exist? (show full working out)
Example 2
Sketch a graph for y = 2 sin (x +) + 1, x € [0,4n] (show full working out for all intercepts and turning
points).
Example 3
Find the derivative of In(3t² – 2t + 1) and then calculate f +3 dt (use integration by recognition).
Example 4
The probability of winning a prize in a game of chance is 0.48. What is the least number of games that must be
played to ensure that the probability of winning at least twice is more than 0.95?
Example 5
The yield of a variety of Farmer Jase's pumpkin patch has a probability density function:
kx, 0 Sx< 3
f(x) = }k(6 – x),3<x56
0,x< 0 or x>6
a. Find the variance
b. Find the probability Pr(u – 1<X<µ + 1)
c. Find the value of a such that Pr(X > a) = 0.6, giving your answer correct to one decimal place.
Example 6
In a small factory producing bamboo products, the standard acceptable measurement for a bamboo straws
lengths are between 10.085 cm and 12.075 cm. Alexis has noticed that 6% of the straws are rejected for being
too short and another 6% are rejected for being too tall. What is the mean and standard deviation of
distribution assuming that the straws' lengths are normally distrībuted? (2dp)
Example 7
A sample of four chocolate bars are randomly selected from a box of ten. Six of the chocolates in the box are
milk chocolate. (without replacement)
a. Create a probability distribution table showing the sampling distribution of the sample proportion
of milk chocolate bars in the sample.
b. Evaluate Pr(0 <P < 0.7) and hence evaluate Pr(P < 0.7 | P > 0).
Example 8
In a particular university, the probability that any one student is male is 0.65. Four students are selected at
random from this university as a sample.
a. Create a probability distribution table showing the sampling distribution of the sample proportion
of female students in the sample. (4dp)
b. Find Pr(P < 0.65 | P > 0). (4dp)
Example 9
A study is done in Sydney to estimate the proportion of people living there who have travelled outside of
Australia. Previous research has found that this proportion is equal to roughly 28%.
a. Find the minimum sample size needed for the study in order to obtain a minimum of 5% margin of
error in an approximate 95% confidence interval for this proportion. (use k=1.96)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c81a207-5a91-4ac9-bdf9-78b1ba278795%2F41714a25-32c6-4589-a315-52c8f8237b3c%2F9p0gftn_processed.png&w=3840&q=75)
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