4. Prove that if f(x) = agx" + a,x"-1 +..+ an-1x = 0 has a positive root x = xo, then the equation na,x"-1 + (n – 1)a,x"-2 + + an-1 %3D = 0 also has a positive root x = A, where A
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Answer question number 4
![purse code and the title, title of the lesson, title of activity, name of student and your block, date of
ubmission, name of your professor). Take note that all pages must have 0.5 inches' border
cluding the front page. Copy the questions then answer. Once you're done answering, send it to me
a Google Classroom in a PDF file. Please avoid erasure. Good Luck
oblem Set No. 6:
1. For the function f(x) = sin x + 2, find the value/s of c given the interval [0, 21).
2. Consider the function f(x) = |x| on the closed interval [-1,1].
3. Let f(x) = x, given the interval [0,1]
4. Prove that if f(x) = agx" + a,x"-1+
equation na,x"-1 + (n – 1)a,x"-2 + ..+ an-1 = 0 also has a positive root x = A, where A<Xo.
5. Prove that the function f(x) = cosx is strictly decreasing on the interval [0,1).
6. For what values of x is the function f(x) = x* – 2x? strictly increasing?
7. Find the intervals for which the function f(x) = x In x hcreasing or decreasing.
+ an-1x = 0 has a positive root x = xa, then the
8. Find the interval on which the function f(x) = x³ + is strictly decreasing.
9. Find the angles at which the curve y =x - x intersects the x-axis.
10. Find the acute angle of intersection of the curves x + xy = 1 and y 3 = (x+ 1)?.
In the 1st quadrant of the curve = 9- x2, a tangent line is drawn. The tangent line intersects
the coordinate axes at points U and V.
11. Find the minimum length of UV.
12. Find the coordinates of the point of tangency.
13. Determine all the number(s) c which satisfy the conclusion of Mean Value Theorem for A(t) =
8t +e-3 on [-2,3].
14. Suppose we know that f(x) is continuous and differentiable on the interval [-7,0], that f(-7)=-3
and that f(x)s2. What is the largest possible value for f(0)?
15. Show that f(x) = x – 7x? +25x +8 has exactly one real root.
16. Suppose that we know that f(x) is continuous and differentiable everywhere. Let's also
suppose that we know that f(x) has two roots. Show that f'(x) must have at least one root.
17. Describe the concavity of the function f(x) = .
18. Describe the concavity of the function f(x) = x? – x
ULE 6 APPLICATION OF DERIVATIVES
GINEERING CALCULUS 1
19. Describe the concavity of the function f(x) = 6x + sin 3x
20. Describe the concavity of the function f(x) = cos?x – sin²x
21. Find any relative extrema of f(x) = x *- 8x² using the Second Derivative Test.
22. Find any relative extrema of f(x) = sin x + cos x on [0,2] using the Second Derivative Test.
23. Let f(0)=cos2(0)- 2sin(e). Find the intervals where f is increasing and the intervals where f is
decreasing in [0,2m). Use this information to classify the critical points of f as either local
maximums, local minimums, or neither.
24. Find all local maximum and minimum points by the second derivative test, when possible. y =
x? +
25. Consider the curve y? = 4 + xand chord AB joining points A(- 4,0) and B(0,2) on the curve.
Find the x coordinate of the point on the curve where the tangent line is parallel to chord AB.
26. The cost of fuel in a locomotive is proportional to the square of the speed and is $25 per hour
for a speed of 25 miles per hour. Other cost amount to $100 per hour regardless of the speed.
What is the speed which will make the cost per mile a minimum?
It is estimated that between the hrs of noon and 7 PM, the speed of a highway traffic flowing
past the intersection of EDSA and Ortigas Avenue is approximately s = t3 - 9t2 + 15t + 45 kph
where't is the number of hours past noon.
27. At what time between noon and 7 PM is the traffic moving the fastest?
28. At what time between noon and 7 PM is the traffic moving the slowest?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F985ce923-3767-45d3-b371-597a007a5504%2Fa50cb3c6-ad57-46d1-9a25-5fe3691619b8%2F7gbty76_processed.jpeg&w=3840&q=75)
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