Expert: Paul Klarreich - 1/5/2010

Question
QUESTION: First question resubmitted slightly wrong.

Determine the equation of the tangent line to the function y=tanx at the point (Pi,0).

Here are 2 other questions if you get the time. Thanks

1. A telephone pole is 10m high. A wire is tethered from the top of the pole to a peg on the ground. Let x be the angle of inclination the wire makes with the ground.
a) Determine an equation that relates the length of the wire,L, to the angle of inclination of the wire.
b) Determine an equation that relates the distance, d, of the peg from the base of the pole to the angle of incination, x.
c) What angle will be made when 15m of wire is used? At what distance will the peg be planted?

2. A weight hangs from a spring attached to the ceiling.  At rest, the spring is 8cm long. A student pulls the weight down, stretching the spring to 15cm, and lets go. The spring reacts to a length of 1 cm in 0.5 seconds. The length of the spring, over time,can be modelled with a sinsoidal function.
a) Sketch the graph of the length of spring versus time over 5 seconds from its release
b) Determine an equation that relates to length of the spring,l,in cm,to the time passed,t,in seconds.
c) determine at what time in the cycle the length of the spring is 12.5cm.

ANSWER: Questioner: Carmine
Country: Canada
Category: Advanced Math
Private: No
Subject: Advanced Functions
Question: First question resubmitted slightly wrong.

Determine the equation of the tangent line to the function y=tanx at the point (Pi,0).

Here are 2 other questions if you get the time. Thanks

1. A telephone pole is 10m high. A wire is tethered from the top of the pole to a peg on the ground. Let x be the angle of inclination the wire makes with the ground.
a) Determine an equation that relates the length of the wire,L, to the angle of inclination of the wire.
b) Determine an equation that relates the distance, d, of the peg from the base of the pole to the angle of incination, x.
c) What angle will be made when 15m of wire is used? At what distance will the peg be planted?

2. A weight hangs from a spring attached to the ceiling.  At rest, the spring is 8cm long. A student pulls the weight down, stretching the spring to 15cm, and lets go. The spring reacts to a length of 1 cm in 0.5 seconds. The length of the spring, over time,can be modelled with a sinsoidal function.
a) Sketch the graph of the length of spring versus time over 5 seconds from its release
b) Determine an equation that relates to length of the spring,l,in cm,to the time passed,t,in seconds.
c) determine at what time in the cycle the length of the spring is 12.5cm.
......................................................

Determine the equation of the tangent line to the function y=tanx at the point (Pi,0).

To find the tangent line:

1. The slope of the tangent line to the curve y = tan x at the point ( PI, 0 ) is just the derivative at  x = pi.

2. Once you have that, you have your m, and use the point-slope form:

y - y0 = m(x - x0)

with x0 = pi,  y0 = 0

...................................

Draw your right triangle, and you have:

x = the angle of inclination,
10 is the leg opposite to x
d is the leg adjacent to x
L is the hypotenuse.

use whatever trig function is appropriate.

..........................................

Since your initial (t = 0) displacement is +7, use a cosine function.

Since the spring will recover its initial position in one second, write

cos (2 pi t)

Since the amplitude is 7, write:

7 cos( 2 pi t)

Since the length is the 'rest' length plus displacement, write:

L = 8 + 7 cos( 2 pi t)

You can handle the rest.


---------- FOLLOW-UP ----------

QUESTION: Thanks Paul
Sorry I got short changed in Trig when I took it so am behind when it comes to trig now. So could you give a complete answer for the tangent question and the spring. If I just sub in 12.5 for t in the last equation you gave me, would that give me the answer. If I use proportional math 1 cm for .5 sec as to 12.5 cm for x. I get 6.25seconds, not what I would get if I plugged in 12.5 into the last formulae. My reasoning may be flawed?

carm

Answer
Questioner: Carmine
Country: United States
Category: Advanced Math
Private: Yes
Subject: Trig functions
Question: QUESTION: First question resubmitted slightly wrong.

Determine the equation of the tangent line to the function y=tanx at the point (Pi,0).

Here are 2 other questions if you get the time. Thanks

1. A telephone pole is 10m high. A wire is tethered from the top of the pole to a peg on the ground. Let x be the angle of inclination the wire makes with the ground.
a) Determine an equation that relates the length of the wire,L, to the angle of inclination of the wire.
b) Determine an equation that relates the distance, d, of the peg from the base of the pole to the angle of incination, x.
c) What angle will be made when 15m of wire is used? At what distance will the peg be planted?

2. A weight hangs from a spring attached to the ceiling.  At rest, the spring is 8cm long. A student pulls the weight down, stretching the spring to 15cm, and lets go. The spring reacts to a length of 1 cm in 0.5 seconds. The length of the spring, over time,can be modelled with a sinsoidal function.
a) Sketch the graph of the length of spring versus time over 5 seconds from its release
b) Determine an equation that relates to length of the spring,l,in cm,to the time passed,t,in seconds.
c) determine at what time in the cycle the length of the spring is 12.5cm.

ANSWER: Questioner: Carmine
Country: Canada
Category: Advanced Math
Private: No
Subject: Advanced Functions
Question: First question resubmitted slightly wrong.

Determine the equation of the tangent line to the function y=tanx at the point (Pi,0).

Here are 2 other questions if you get the time. Thanks



......................................................

Determine the equation of the tangent line to the function y=tanx at the point (Pi,0).

To find the tangent line:

1. The slope of the tangent line to the curve y = tan x at the point ( PI, 0 ) is just the derivative at  x = pi.

dy/dx = sec^2 x,  at  x = pi, dy/dx = 1/(cos pi)^2 = 1 = m


2. Once you have that, you have your m, and use the point-slope form:

y - y0 = m(x - x0)

with x0 = pi,  y0 = 0

y - 0 = 1(x - pi)

y = x - pi

(you really needed me for that?)

...................................

1. A telephone pole is 10m high. A wire is tethered from the top of the pole to a peg on the ground. Let x be the angle of inclination the wire makes with the ground.
a) Determine an equation that relates the length of the wire,L, to the angle of inclination of the wire.
b) Determine an equation that relates the distance, d, of the peg from the base of the pole to the angle of incination, x.
c) What angle will be made when 15m of wire is used? At what distance will the peg be planted?

Draw your right triangle, and you have:

x = the angle of inclination,
10 is the leg opposite to x
d is the leg adjacent to x
L is the hypotenuse.

use whatever trig function is appropriate.

a) sin x = 10/L

b) tan x = 10/d

c) If L = 15,  sin x = 10/15 = 0.6666.  Now use your calculator to find inverse-sin of 0.6666

If L = 15, use the Pythagorean Theorem (you

..........................................

2. A weight hangs from a spring attached to the ceiling.  At rest, the spring is 8cm long. A student pulls the weight down, stretching the spring to 15cm, and lets go. The spring reacts to a length of 1 cm in 0.5 seconds. The length of the spring, over time,can be modelled with a sinsoidal function.
a) Sketch the graph of the length of spring versus time over 5 seconds from its release
b) Determine an equation that relates to length of the spring,l,in cm,to the time passed,t,in seconds.
c) determine at what time in the cycle the length of the spring is 12.5cm.
.....................
Since your initial (t = 0) displacement is +7, use a cosine function.

Since the spring will recover its initial position in one second, write

cos (2 pi t)

Since the amplitude is 7, write:

7 cos( 2 pi t)

Since the length is the 'rest' length plus displacement, write:

L = 8 + 7 cos( 2 pi t)

a) Your graph should show 5 full cycles.

c) Set L = 12.5:

12.5 = 8 + 7 cos( 2 pi t)

4.5 =  7 cos( 2 pi t)
cos( 2 pi t) = 4.5/7

OK, now use your calculator like this:

Click Radians button.
4.5
/
7
=
INV
cos
/
2
/
pi
=

I got 0.10231389198764700156571099073947 seconds.

---------- FOLLOW-UP ----------

QUESTION: Thanks Paul
Sorry I got short changed in Trig

>> yes, that happens often when the professor does not pace the algebra-trig or precalculus course and finds he has only 2 hours left to teach all of trig.

NO, IT NEVER HAPPENED TO ME.

........................

when I took it so am behind when it comes to trig now. So could you give a complete answer for the tangent question and the spring. If I just sub in 12.5 for t in the last equation you gave me, would that give me the answer. If I use proportional math 1 cm for .5 sec as to 12.5 cm for x. I get 6.25seconds, not what I would get if I plugged in 12.5 into the last formulae. My reasoning may be flawed?

..............
A bit, yes.