Math Answers

Activity in Limit Theorems

Compute the following limits.

1. lim (4 • f(x))

x→c

2. lim (g(x) - h (x))

x→c ________

3. lim √12 • f(x)

x→c

4. lim (g(x) + h(x)) / f(x)

x→c

5. lim (f(x) + h(x))

x→c

Activity in Limit Theorems

Directions: Assume the following.

lim f(x) = 3/4;

x→c

lim g(x) = 12;

x→c

lim h(x) = -3;

x→c

3. An ideal shock absorption system would use a critically damped oscillator to absorb shock loads. The location of the absorbing piston (𝑥) is described by 𝑥 = 𝜏𝑒−𝛾𝑡 where:

- 𝜏 is the linear damping coefficient

- 𝛾 is the exponential damping constant

- 𝑡 is the time (𝑠)

- 𝑥 is the displacement of piston (𝑚)

The tasks are to:

a) Draw a graph of displacement against time for 𝜏 = 12 and 𝛾 = 2, between 𝑡 = 0𝑠 and 𝑡 = 10𝑠.

b) Calculate the gradient at 𝑡 = 2𝑠 and 𝑡 = 4𝑠.

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   c) Differentiate the function of 𝑥 and calculate the value of 𝑑𝑥 at 𝑡 = 2𝑠 and 𝑡 = 4𝑠. 𝑑𝑡

d) Compare your answers for part b and part c. (M1)

e) Calculate the derivative for the velocity function(𝑑2𝑥).

Determine whether if

lim f(c) = f(c)

x→c

1. f(x) = x+2; c = -1

2. f(x) = x-2; c = 0

3. (at c = -1 )

f(x) = {x ² - 1 if x < -1}

f(x) = { (x - 1) ² - 4 if x ≥ -1}

4. (at c = 1 )

f(x) = {x³ - 1 if x < 1}

f(x) = { x² + 4 if x ≥ 1}

To ensure a growing season of sufficient length, Mang Popoy has at most 16 days left to

plant corn and soybean . he can plant com at a rate of 10 hectares per day and soybean at

15 hectares per day. If he has at most 200 hectares available, how many hectares of each

type of crop can he plant?​

suppose that the average outstanding credit card balance for young professionals is Php 11,200 with standard deviation of Php 2,600. In simple random sample of 150 young professionals, what is the probability that the mean outstanding credit card balance does not exceeds Php 12, 300?

Q.You have been hired as a researcher for a prominent research-based organization and you have been allocated a new project for helping out a client.

You know that the time taken to complete similar projects is normally distributed with a mean of 50 days and a standard deviation of 4 days.  

a. What is the probability of finishing this project within 45 – 55 days? (5 Marks) 

b. When do we use Poisson Distribution? Give two example scenarios. (5 Marks)

Consider the following sample. It contains the weights of 25 participants:

55, 57, 58, 43, 62, 67, 71, 69, 66, 51, 72, 62.5, 58.5, 61, 72.5, 75, 44, 48, 49, 49.5, 49, 62, 66, 71, 58.

Calculate and plot the following using MS-Excel.

Write the formula and show the steps.

a. Variance 

b. Standard deviation

c. Histogram

d. Ogive

Note: For histogram and ogive use bins numbers as 40, 45, 50, 55, 60, 65, 70     

A health scientist has created three different diet plans for a weight loss program. He conducted an experiment with 24 volunteers to see the impact on weight loss. He measured the weights of the volunteers before the start of the diet plan. Then they were assigned to one specific diet plan for a month. After one month their weights were measured again. The differences in weights are given in the table below for each of the diet plans. Read data table and answer the questions.

Table (3Rows/8 columns)

DietPlan1|DietPlan2|DietPlan3

2 -2.1 7

8.5 2 5.6

1.9 1.7 3.4

3.1 4.3 6.8

1.5 7 7.8

3 0.6 5.4

0.9 2.7 6.8

2.8 2 5

a. Conduct ANOVA on the above table. Do you find a significant difference in weight loss due to the three diet plans?

Explain and interpret using both the p-value and F-critical value from the output. Use MS-Excel for the analysis.

b. Why do you think ANOVA is required to answer this question?

A system vibrates according to the equation x ''(t)+9x(t)=6sin3t , where x is the displacement and t is the time. Determine a general solution for x(t) by using the method of undetermined coefficients .