# Am I Right? (Differentiation)

• Oct 25th 2013, 03:26 PM
Fratricide
Am I Right? (Differentiation)
1. Find the equation of the tangent at the given points for: f(x) = x2 + 3x, (2, 10)
I did:
f'(x) = 2x + 3
Let x = 2
--> m = 7
y - 10 = 7(x - 2)
y = 7x - 4

2. Find the equation of the tangent to the curve y = x3 - 9x2 + 20x - 8 at the point (1, 4). At what points of the curve is the tangent parallel to the line y = -4x + 3?
First, I used the method from the previous question to obtain the equation for the parallel line. For the next part of the question, I equated the gradient function to the gradient of the parallel line and solved for x, substituting the newly found x values into the original curve equation to get my y-coordinates.

3. A model plane flying level at 250 m above the ground suddenly dives... blahablahhal... h(t) = 8t2 - 80t + 250. Find the rate at which the plane is losing height at t = 3.
I simply substituted t = 3 into the original equation and solved, giving my answer as 82 ms-1. Is this question as simple as that?

4. A car starts from rest and moves a distance s (m) in t (s), where s = (1/6)t3 + (1/4)t2. What is the acceleration when t = 2?
I know that velocity = ds/dt and acceleration = dv/dt, so I found that s' = (1/2)t2 + (1/2)t and v' = t + (1/2). Then, to answer the question, I simply substituted t = 2 into the equation for acceleration and solved, giving my (5/2) ms-2.

Thanks in advance. Your feedback is greatly appreciated.
• Oct 25th 2013, 05:50 PM
HallsofIvy
Re: Am I Right? (Differentiation)
Quote:

Originally Posted by Fratricide
1. Find the equation of the tangent at the given points for: f(x) = x2 + 3x, (2, 10)
I did:
f'(x) = 2x + 3
Let x = 2
--> m = 7
y - 10 = 7(x - 2)
y = 7x - 4

Yes,, that is correct.

Quote:

2. Find the equation of the tangent to the curve y = x3 - 9x2 + 20x - 8 at the point (1, 4). At what points of the curve is the tangent parallel to the line y = -4x + 3?
First, I used the method from the previous question to obtain the equation for the parallel line. For the next part of the question, I equated the gradient function to the gradient of the parallel line and solved for x, substituting the newly found x values into the original curve equation to get my y-coordinates.
Sounds good. Why have you not done it?

Quote:

3. A model plane flying level at 250 m above the ground suddenly dives... blahablahhal... h(t) = 8t2 - 80t + 250. Find the rate at which the plane is losing height at t = 3.
I simply substituted t = 3 into the original equation and solved, giving my answer as 82 ms-1. Is this question as simple as that?
No, it's not. Putting h= 3 into h(t) gives h(3)= 82 meters- not "meters per second", because, as you were told, h(t) is the height of the plane at t seconds, NOT the rate at which that height is changing.

Quote:

4. A car starts from rest and moves a distance s (m) in t (s), where s = (1/6)t3 + (1/4)t2. What is the acceleration when t = 2?
I know that velocity = ds/dt and acceleration = dv/dt, so I found that s' = (1/2)t2 + (1/2)t and v' = t + (1/2). Then, to answer the question, I simply substituted t = 2 into the equation for acceleration and solved, giving my (5/2) ms-2.
Yes, that is correct. Why didn't you do that for the previous question?

Quote:

Thanks in advance. Your feedback is greatly appreciated.
• Oct 25th 2013, 06:24 PM
Fratricide
Re: Am I Right? (Differentiation)
Quote:

Originally Posted by HallsofIvy
Sounds good. Why have you not done it?

I have, I just didn't include it for brevity's sake.
Here it is:
dy/dx = 3x2 - 18x + 20
Let x = 1
--> m = 5
y - 4 = 5(x - 1)
y = 5x - 1

y = -4x + 3
m = -4
Let 3x2 - 18x + 20 = -4
3x2 - 18x + 24 = 0
(3x - 12)(x - 2) = 0
x = 2, 4

Let x = 2
y = 2(2)3 - 9(2)2 + 20(2) - 8
y = 4

Let x = 4
y = 2(4)3 - 9(4)2 + 20(4) - 8
y = -8

--> Tangent is parallel to the line y = -4x + 3 at points (2,4) and (4,-8).

Quote:

Originally Posted by HallsofIvy
Why didn't you do that for the previous question?

Ahhhh, I see. So I need to use the same method as in 4, substituting t = 3 into v'?
• Oct 26th 2013, 03:14 PM
Fratricide
Re: Am I Right? (Differentiation)
Ignore that last part of my previous post. I'm pretty sure that for question three I need to find the velocity (not acceleration) at the given time.
As such, I did:
h'(t) = 16t - 80
Let t = 3
h'(t) = -32 ms-1

Am I on the right track? (With this question and the workings out in my previous post.)
• Oct 28th 2013, 12:31 PM
Fratricide
Re: Am I Right? (Differentiation)
I'm pretty sure what I've done is correct, but can anyone confirm it for me?
• Oct 28th 2013, 03:21 PM
Prove It
Re: Am I Right? (Differentiation)
If you get h'(3) = -32, then the plane is descending at 32m/s.