# I need help on a few things

• Jun 20th 2008, 11:48 AM
wren17
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.
• Jun 20th 2008, 11:59 AM
Mathstud28
Quote:

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
• Jun 20th 2008, 12:09 PM
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.
• Jun 20th 2008, 12:13 PM
Mathstud28
Quote:

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.

• Jun 20th 2008, 12:21 PM
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.
• Jun 20th 2008, 12:22 PM
Mathstud28
Quote:

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?
• Jun 20th 2008, 12:26 PM
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.
• Jun 20th 2008, 12:32 PM
Mathstud28
Quote:

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.