# Thread: I need help on a few things

1. ## I need help on a few things

1. The limit, as x tends to 2, of the function $y=\frac{x^2 + bx - 28}{x - 2}$ exists, and is equal to 16. What is the value of the whole number coefficient b?

2. The first derivative of $y = (6x^3 + 11)^9$ is of the form $ax^2(6x^3 + 11)^n$ , where a is a positive number. What is the value of the whole number coefficient a?

3. For the ellipse $4x^2 + 9y^2 = 724$, the slope of the tangent at the point (10, -6) is b. Correct to the nearest hundredth, the value of b is ______________.

4. The number I is defined as $\int_{1}^{2} 4~sin~x~cos~xdx$. The value of I, correct to the nearest hundredth, is ______________.

5. A spring has a motion that can be modelled by the differential equation $\frac{d^2 y}{dt^2} = -5y$. One solution of this equation is $y = 4~cos~kt$, where k is a positive number to be determined. What is the value of k, correct to two decimal places?

6. An LRT train travels for 120 s between two stations. It accelerates for 30 s, maintains a constant velocity for 70 s, and brakes to a stop in 20 s. The velocity function v(t), with velocity measured in m/s, is defined as follows:
$v(t) = 20sin[\frac{\pi t}{60}]~~~~~~~~0 \geq t \geq 30$
$v(t) = 20 ~~~~~~~30 \geq t \geq 100$
$v(t) = -0.05(t - 100)^2 + 20~~~~~~~~100 \geq t \geq 120$

a. Sketch a graph of the velocity as a function of time, assuming the velocity function to be continuous at t = 30 and t = 100. (Be sure to label your axes.)

b. Determine the total distance travelled by the train in the 120-second time interval.

4. Let the function f be defined as $f(x) = \frac{tan~x - sin~x}{x^3}$.
a. Use a suitable numerical value of x to estimate the limit, if it exists, of f(x) as x approaches zero.
b. Use trigonometric identities and limit theorems to confirm the estimate made in part a above.

2. Originally Posted by wren17
1. The limit, as x tends to 2, of the function $y=\frac{x^2 + bx - 28}{x - 2}$ exists, and is equal to 16. What is the value of the whole number coefficient b?

2. The first derivative of $y = (6x^3 + 11)^9$ is of the form $ax^2(6x^3 + 11)^n$ , where a is a positive number. What is the value of the whole number coefficient a?

3. For the ellipse $4x^2 + 9y^2 = 724$, the slope of the tangent at the point (10, -6) is b. Correct to the nearest hundredth, the value of b is ______________.

4. The number I is defined as $\int_{1}^{2} 4~sin~x~cos~xdx$. The value of I, correct to the nearest hundredth, is ______________.

5. A spring has a motion that can be modelled by the differential equation $\frac{d^2 y}{dt^2} = -5y$. One solution of this equation is $y = 4~cos~kt$, where k is a positive number to be determined. What is the value of k, correct to two decimal places?

6. An LRT train travels for 120 s between two stations. It accelerates for 30 s, maintains a constant velocity for 70 s, and brakes to a stop in 20 s. The velocity function v(t), with velocity measured in m/s, is defined as follows:
$v(t) = 20sin[\frac{\pi t}{60}]~~~~~~~~0 \geq t \geq 30$
$v(t) = 20 ~~~~~~~30 \geq t \geq 100$
$v(t) = -0.05(t - 100)^2 + 20~~~~~~~~100 \geq t \geq 120$

a. Sketch a graph of the velocity as a function of time, assuming the velocity function to be continuous at t = 30 and t = 100. (Be sure to label your axes.)

b. Determine the total distance travelled by the train in the 120-second time interval.

4. Let the function f be defined as $f(x) = \frac{tan~x - sin~x}{x^3}$.
a. Use a suitable numerical value of x to estimate the limit, if it exists, of f(x) as x approaches zero.
b. Use trigonometric identities and limit theorems to confirm the estimate made in part a above.
1)For the first one, there are two choices, the limit does not exist, or it is indeterminate (i.e. the top has a factor of x-2). And since the limit exists the latter must be true. Your choices are divide, or use L'hopital's. Either way once you have simplified this is just a simple matter of subbing and and solving for b

2)Use chain rule

$y'=9(6x^3+11)^8\cdot(6x^3+11)'$

Then visually inspect and equate coefficients

3) Your best bet here is to just multiply through by 36, and then use implicit differentiation

4) $I=4\int_1^{2}\sin(x)\cos(x)dx$

You can decimate this one by noting that

$\sin(2x)=2\sin(x)\cos(x)\Rightarrow\sin(x)\cos(x)= \frac{1}{2}\sin(2x)]$

5) This is an SDE (Seperable Differential Equation), seperate variables and integrate. If you have any problems report back

6) Here, you must just sketch it, and remember that

$\text{Total Distance}=\int_a^{b}|v(t)|dt$

where || denotes absolute value

7) I am assuming that this is just the limit as x goes to zero?

For the approximation put in x=.00001 and x=-.00001

For the actual limit, I assume you do not know Maclaurin series, so I would suggest L'hopital's. Or a clever usage of algaebraic manipulation.

If you have any problems just post back

Mathstud

3. Thanks for the help, but is it possible to get the actual solutions to these, these were just some examples that i didn't get the solutions for and i still have more similar questions like these that i need to do i just wanted to have these to compare to.

4. Originally Posted by wren17
Thanks for the help, but is it possible to get the actual solutions to these, these were just some examples that i didn't get the solutions for and i still have more similar questions like these that i need to do i just wanted to have these to compare to.
I will help you get the final solutions, if you post your work unto now.

5. Well i haven't done any work for these yet, i was working on other problems which i had more knowledge on. These were the ones i wasn't quiet as familiar with. Thats why i posted them here because i don't feel ill be able to complete these ones accurately and i still have more similar to these to do.

6. Originally Posted by wren17
Well i haven't done any work for these yet, i was working on other problems which i had more knowledge on. These were the ones i wasn't quiet as familiar with. Thats why i posted them here because i don't feel ill be able to complete these ones accurately and i still have more similar to these to do.
Well, I gave you a start for each. So why not work on them a little and when, if you do, get caught come back and we will work on them together?

7. Ok ill try that, thanks for the help, i doubt ill be able to finsh them today though since im going out of town in a few hours.

8. Originally Posted by wren17
Ok ill try that, thanks for the help, i doubt ill be able to finsh them today though since im going out of town in a few hours.
Well, whenver you get a chance to post them, I am sure someone on here will be glad to help.