Question

Instant fan/medal:

Find the exact value of the slope of the line which is tangent to the curve given by the equation
r = 1 + cos θ at pi/2=theta.

Answer 1

find the gradient first

Answer 2

Do you have the expression for gradient?

Answer 3

@FaiqRaees I don't even know what a gradient is unfortunately, my online course is pretty vague. This is my first time working with polar curves :)

Answer 4

Do you know differentiation?

Answer 5

I am not able to help you if you wont respond.

Answer 6

@FaiqRaees yes I do know differentiation, but I am not sure how complex the problem is yet

Answer 7

Okay so to differentiate y
First differentiate 1
Whats the answer?

Answer 8

d/dx of 1 = 0?

Answer 9

Perfect
Now differentiate cos θ

Answer 10

-sin(theta)

Answer 11

Very good
d/dx y = 0+ (-sin θ)
d/dx y = -sin θ

Answer 12

Do you understand how we arrived at the concluded gradient?

Answer 13

yes we derived the polar curve

Answer 14

Okay now we have to find the gradient at θ=π/2 so substitute θ=π/2 in the equation of gradient and find out the gradient at θ=π/2.

Answer 15

-sin(pi/2)=-1

Answer 16

Yes very good. Now we have the gradient of our tangent. All we need is a point which lies on the tangent. Any idea how can we figure out that point?

Answer 17

graph?

Answer 18

We can do that but its complicated. What we will do is, we know that tangent touches the curve at θ=π/2. Which means the coordinate at θ=π/2 will lie both on the curve and the tangent. Agree?

Answer 19

Yep :)

Answer 20

So substitute θ=π/2 in the equation of curve to find the y coordinate at θ=π/2

Answer 21

1+cos(pi/2)=1

Answer 22

so our coordinates are (π/2,1)
Now substitute the coordinates in the equation
y= -x+c (where c is a constant to be found)

Answer 23

Confused?

Answer 24

1=-pi/2 +c, c=1+(pi/2) ? @FaiqRaees sorry for slow response there

Answer 25

yes now substitute the value of c in y= -x+c

Answer 26

y=-x+1+pi/2

Answer 27

yes correct, there you have the equation of tangent

Answer 28

So how do I go about finding slope now?

Answer 29

Oh btw I am really really sorry, I misread the questio, the question was way shorter. After finding the derivative, you just had to plug θ=π/2 and you would have your answer -1

Answer 30

lol it's fine ! Thank you for everything @FaiqRaees

Answer 31

dy/dx = -sinθ
dy/dx = -sin(π/2)
dy/dx = -1
Slope =-1