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Left Hand, Right Hand Derivatives and Parametric Curves Worksheet

Left Hand, Right Hand Derivatives and Parametric Curves Worksheet
  • Page 1
 1.  
Find dydx for the parametric curve x = 6t, y = 10t2.
a.
10 3t
b.
t
c.
1t
d.
3 10(1t)
e.
1


Solution:

x = 6t , y = 10t2
[Equations of the parametric curves.]

dxdt = 6
[Differentiate x with respect to t.]

dydt = 20t
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

= 20t6 = 10 / 3t

dydx = 10 / 3t


Correct answer : (1)
 2.  
Find dydx for the parametric curve x = 5t2 - 6, y = t + 11.
a.
110
b.
110t
c.
10t
d.
1t
e.
t


Solution:

x = 5t2 - 6, y = t + 11
[Equations of the parametric curves.]

dxdt = 10t
[Differentiate x with respect to t.]

dydt = 1
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

= 110t

dydx = 110t


Correct answer : (2)
 3.  
Find the slope of the parametric curve x = 7cos t, y = 7sin t.
a.
7cos t
b.
cot t
c.
- tan t
d.
- cot t
e.
tan t


Solution:

x = 7cos t , y = 7sin t
[Equations of the parametric curves.]

dxdt = - 7sin t
[ Differentiate x with respect to t .]

dydt = 7cos t
[Differentiate y with respect to t .]

dydx = dydtdxdt
[Find slope of the curve, dydx.]

= 7cos t- 7sint = - cot t
[Simplify.]

Slope of the parametric curve is - cot t


Correct answer : (4)
 4.  
Find dydx for the parametric curve x = 2e8t, y = te8t.
a.
18e8t
b.
e8t
c.
e8t(1+8t)
d.
1 + 8t
e.
1+8t16


Solution:

x = 2e8t, y = te8t
[Equations of the parametric curves.]

dxdt = 16e8t
[Differentiate x with respect to t.]

dydt = e8t + 8te8t
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

dydx = e8t+8te8t16e8t

= 1+8t16
[Simplify.]

dydx = 1+8t16


Correct answer : (5)
 5.  
Find dydx for the parametric curve x = 2t, y = t - 48.
a.
1t2
b.
2
c.
- 1t2
d.
t2
e.
- 1 2 t2


Solution:

x = 2t, y = t - 48
[Equations of the parametric curves.]

dxdt = - 2t2
[Differentiate x with respect to t.]

dydt = 1
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

= 1- 2t2 = - 1 / 2 t2
[Simplify.]

dydx = - 1 / 2 t2


Correct answer : (5)
 6.  
Find the slope of the parametric curve x = t3 - 8t, y = 5t2.
a.
2t3t-1
b.
13(t2-1)
c.
tt2-1
d.
2t
e.
10t3t2-8


Solution:

x = t3 - 8t, y = 5t2
[Equations of parametric curves.]

dxdt = 3t2 - 8
[Differentiate x with respect to t.]

dydt = 10t
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find slope of the curve, dydx.]

= 10t3t2-8

The slope of the parametric curve is 10t3t2-8


Correct answer : (5)
 7.  
Find the slope of the parametric curve x = asec 8t, y = btan 8t.
a.
abcos 8t
b.
absin 8t
c.
batan 8t
d.
basec 8t
e.
bacosec 8t


Solution:

x = asec 8t , y = btan 8t
[Equations of the parametric curves.]

dxdt = 8asec 8t tan 8t
[ Differentiate x with respect to t .]

dydt = 8bsec2 8t
[Differentiate y with respect to t .]

dydx = dydtdxdt
[Find slope of the curve, dydx.]

= bsec2 8ta(sec 8t)(tan 8t) = bacosec 8t
[Simplify.]

Slope of the parametric curve is bacosec 8t


Correct answer : (5)
 8.  
Find dydx for the parametric curve x = 9 - t2, y = 7-t2.
a.
27-t2
b.
t27-t2
c.
17-t2
d.
127-t2
e.
- 127-t2


Solution:

x = 9 - t2, y = 7-t2
[Equations of the parametric curves.]

dxdt = - 2t
[Differentiate x with respect to t.]

dydt = - t7-t2
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

= - t7-t2- 2t

= 127-t2
[Simplify.]

dydx = 127-t2


Correct answer : (4)
 9.  
Find the slope of the parametric curve x = cos3 2t, y = sin3 2t.
a.
- tan 2t
b.
6sin2 2tcos 2t
c.
- cot 2t
d.
tan 2tsin 2t
e.
- tan 2tsin 2t


Solution:

x = cos3 2t , y = sin3 2t
[Equations of the parametric curves.]

dxdt = - 6cos2 2t sin 2t
[ Differentiate x with respect to t .]

dydt = 6sin2 2t cos 2t
[Differentiate y with respect to t .]

dydx = dydtdxdt
[Find slope of the curve, dydx.]

= 6sin2 2t cos 2t- 6cos2 2t sin 2t = - tan 2t
[Simplify.]

Slope of the parametric curve is - tan 2t


Correct answer : (1)
 10.  
Find dydx for the parametric curve x = e7tcos 9t, y = e7tsin 10t.
a.
7sin 10t+10cos 10t7cos 9t-9sin 9t
b.
7cos 9t-9sin 9t7cos 10t+10sin 10t
c.
- cot 9t
d.
1cos 9t-sin 9t
e.
tan 10t


Solution:

x = e7tcos 9t, y = e7tsin 10t
[Equations of the parametric curves.]

dxdt = e7t(7cos 9t - 9sin 9t)
[Differentiate x with respect to t.]

dydt = e7t(7sin 10t + 10cos 10t)
[Differentiate y with respect to t.]

dydx = dydtdxdt
[Find dydx.]

= e7t(7sin 10t+10cos 10t)e7t(7cos 9t-9sin 9t)

= 7sin 10t+10cos 10t7cos 9t-9sin 9t
[Simplify.]

dydx = 7cos 10t+10sin 10t7cos 9t-9sin 9t


Correct answer : (1)

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