Section 2

Parametric Differentiation

TheoremLet x = f(t), y = g(t) and dx/dt is nonzero, then

dy/dx = (dy/dt) / (dx/dt)

; provided the given derivatives exist

Example 1Let x = 4sint, y = 3cost.

Find:1. dy/dx and d2y/dx2

2. dy/dx and d2y/dx2

at t = π/43. Find the slope & the equation of the

tangent at t = π/4

Solution- Part1dx/dt = 4cost, dy/dt = -3sint→ dy/dx = (dy/dt) / (dx/dt) = - 3sint / 4cost = -(3/4)tant

d(dy/dx)/dt = -(3/4)sec2td2y/dx2 = d(dy/dx )/dx= [d(dy/dx)/dt] / [dx/dt] = [-(3/4)sec2t ] / 4cost = (-3/16) sec3t

(dy/dx)( π/4) = -(3/4)tan( π/4) = -(3/4)(d2y/dx2 )( π/4) = (-3/16) sec3( π/4) = -(3/16)(√2) 3 = - 3√2/8

Part 2The slope of the tangent at t = π/4 is equal to the

derivative dy/dy at t = π/4, which is-(3/4).The Cartesian coordinates of the point t = π/4 are:x = 4sint π/4 = 4(1/√2)= 2 √2 and y = 3cos π/4= 3(1/√2)= (3/2) √2The equation of the tangent at t = π/4 is:y -(3/2) √2 = -(3/4) ( x - 2 √2 )

Example 2

Let x = 4sint, y = 3cost.

Find:

1.dy/dx and d2y/dx2 at the point (0, -3 )2.The equation of the tangent to the curve at

that point.

Solution – Part 1

. Let x = 4sint, y = 3cost.First we find any value of t corresponding to the

point (0, -3 ). It is clear that one such value is t = π. Why?*Now, we substitute that in the formulas of dy/dxand d2y/dx2 , which we have already deduced in

the Example(1) (dy/dx)( π) = -(3/4)tan( π) = 0

(d2y/dx2 )( π) = (-3/16) sec3( π) = -(3/16)(-1) 3 = 3/16

*Answering “Why?”We have: 0 = 4sint, -3 = 3cost. → sint = 0 & cost = -1 → t = ………,-3π, -π, π, 3π, 5π,…..Notice that the values for any trigonometric function at any of these numbers (angles) are the same. Take: t = π

Solution – Part 2

From the slope of the tangent at (0 ,-3 ) it is clear that the tangent is horizontal, and hence it’s equation is: y = -3

We could also get that from the straight line’s formula:

y – (-3)= ( x -0 )→ y + 3 =0→ y = - 3But this is not very smart. It is like catching a fly

with a hammer!

Book Example (1)

Let x =t2, y = t3 - 3t.

1. Find the equations of all tangents at (3,0)

2. Determine at which point (points), the graph has a horizontal tangent.

3. . Determine at which point (points), the graph has a vertical tangent.

*4. . Determine when the curve is concave upward /

concave downward

Solution-Part 1We have:dx/dt = 2t , dy/dt = 3t2 -3dy/dx = (dy/dt) / (dx/dt) = (3t2 -3) / t

When x =3 & y=0 → 3=t2→ t=√3 or t= -√3

At t=√3, we have dy/dx=(9-3)/(2√3 )= √3 At t= -√3, we have dy/dx=(9-3)/(-2√3 )= -√3

Thus, the equations of the tangents to the curve at (x,y) = (3,0) are: y - 0 = √3 (x - 3) & y - 0 = -√3 (x - 3)That’s: y = √3 (x - 3) & y = -√3 (x - 3)

Solution-Part 2We have: x =t2, y = t3 - 3tdx/dt = 2t , dy/dt = 3t2 -3dy/dx = (dy/dt) / (dx/dt) = (3t2 -3) / 2t

The curve has horizontal tangent when dy/dt = 3t2 -3 =0, while dx/dt = 2t ≠ 0 → t = 1 or t = -1 Why?

At t =1 →x=(1) 2=1 & y = (1)3– 3(1) = 1 - 3 = -2At t =-1 →x=(-1) 2=1 & y = (-1)3 – 3(-1) = -1 + 3 = 2Thus the curve has horizontal tangent at the points:(1,2) & (1,-2)

What’s the equations of these tangents?

Solution-Part 3We have: x =t2, y = t3 - 3tdx/dt = 2t , dy/dt = 3t2 -3dy/dx = (dy/dt) / (dx/dt) = (3t2 -3) / 2t

The curve has vertical tangent when dy/dt = 3t2 -3 ≠ 0, while dx/dt = 2t = 0 → t = 0 Why?

At t =0 →x=(0) 2=0 & y = (0)3– 3(0) = 0

Thus the curve has vertical tangent at the point (0,0)What’s the equations of this tangent?

Solution-4**We have:dx/dt = 2t , dy/dt = 3t2 -3dy/dx = (dy/dt) / (dx/dt) = (3t2 -3) / 2t = (3/2)t – (3/2)t-1

d2y/dx2 = d(dy/dx )/dx= [d(dy/dx)/dt] / [dx/dt]= [(3/2) +(3/2)t-2 ] / 2t = (3/4) t-1 + (3/4)t-3 = [3t2 + 3] / 4t3

d2y/dx2 > 0 if t > 0 & d2y/dx2 < 0 if t < 0 Thus the curve is concave upward if t > 0 and downward if t < 0 We had: x =t2, y = t3 - 3t (t > 0 on the first quadrant. Why? and t

< 0 on the fourth quadrant. Why?

→ ,→

Book Example (2)Let x =r(t - sint), y =r(1 - cost), Where r is a

constant

1. Find the slope of the tangent at t = π/3

2. Determine at which point (points), the graph has a horizontal tangent.

3. . Determine at which point (points), the graph has a vertical tangent

Solution-Part1We have:x =r(t - sint), y =r(1 - cost),dx/dt = r(1 - cost), , dy/dt = rsint→ dy/dx = (dy/dt) / (dx/dt) = rsint / r(1 - cost) = sint / (1 - cost)

(dy/dx)( π/3) = sin( π/3) / [1 - cos( π/3)] = (√3/2)/[1-(1/2)] = √3

Solution-Part 2We have:x =r(t - sint), y =r(1 - cost),dx/dt = r(1 - cost), , dy/dt = rsint→ dy/dx = (dy/dt) / (dx/dt) = rsint / r(1 - cost) = sint / (1 - cost)

The curve has horizontal tangent when dy/dt = rsint=0, while dx/dt = r(1 - cost), ≠ 0 → t = nπ and t ≠ 2nπ

Why?→ t=(2n-1) π ; n is an integer.→x =r[(2n-1) π - 0)] = r(2n-1) π , y =r(1 – (-1) = 2r,

Thus the curve has horizontal tangent at the points:(r(2n-1) π ,2r) Examples: ….,(-3rπ ,2r), (-rπ ,2r), (rπ ,2r), (3rπ ,2r), …

Solution-Part 3We have:x =r(t - sint), y =r(1 - cost),dx/dt = r(1 - cost), , dy/dt = rsint→ dy/dx = (dy/dt) / (dx/dt) = rsint / r(1 - cost) = sint / (1 - cost)

r(1 - cost), = 0 → t = 2nπ ; n is an integer Why?→x =r[(2nπ - 0)] = 2rnπ , y =r(1 – (1) = 0

Checking: show that dy/dx → + ∞ as t → 2nπ from theright*Thus the curve has vertical tangent at the points:(2nrπ , 0)Examples:…, (-4rπ , 0), (-2rπ , 0), (0, 0), (2rπ , 0), (4π , 0),

….

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