The ”Witch” of Agnesi
Find all points on the graph of the function
f (x) =
8a2
x 2 + 4a2
at which the tangent line is horizontal.
Figure: Maria Agnesi, the first person to write a book on differential and integral
calculus. Studied the curve, or ”witch,” of Agnesi
MATH 125
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The University of Kansas
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The Witch of Agnesi in parametric form
dy
Given x(t) = 22a cot(t) and y (t) = a[1 − cos(2t)], find
by first finding
dx
dy
dx
and
dt
dt
Opportunities for Women in Mathematics
•https://sites.google.com/site/awmmath/info/undergrads
•https://www.usnews.com/news/blogs/stem-education/2011/12/
13/9-college-scholarships-for-women-in-stem
MATH 125
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Exercise 7 from Last Week:
Find the derivatives of the
following functions:
(a) y = [x + (x + sin(2x))6 ]7
(b) H(x) = cos(37 + x 7 )
(c) y = sin(tan(8x))
MATH 125
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The University of Kansas
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Exercise 7:
Find the derivatives of the following functions:
(a) y = [x + (x + sin(2x))6 ]7
Solution:
y 0 = 7[x + (x + sin(2x))6 ]6 (1 + 6(x + sin(2x))5 (1 + 2 cos(2x)))
(b) H(x) = cos(37 + x 7 )
Solution:
H 0 (x) = sin(37 + x 7 )(7x 6 )
(c) y = sin(tan(8x))
Solution:
y 0 = cos(tan(8x))8 sec2 (8x)
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The University of Kansas
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Exercise 9:
Solve the following using the new limit identities.
(a) lim
sin(6x)
cos(9x)
(b) lim
tan(16t)
sin(4t)
x→0
t→0
cos(7θ) − 1
θ→0
sin(9θ)
(c) lim
MATH 125
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The University of Kansas
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Exercise 9 Solution:
sin(6x)
x→0 cos(9x)
(a) lim
=
sin(0)
=0
cos(0)
tan(16t)
t→0 sin(4t)
(b) lim
sin(16t)
1
·
t→0 cos(16t) sin(4t)
= lim
1
sin(16t) 4(4t)
·
·
=4
t→0 cos(16t)
16t
sin(4t)
= lim
(c) lim
θ→0
cos(7θ) − 1
sin(9θ)
−(1 − cos(7θ))
9θ
7θ
·
·
=0
θ→0
7θ
sin(9θ) 9θ
= lim
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Exercise 10:
Find the equation of both tangent lines to the ellipse
x 2 + 9y 2 = 81 that pass through the point (27, 3)
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The University of Kansas
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Exercise 10:
Find the equation of both tangent lines to the ellipse
x 2 + 9y 2 = 81 that pass through the point (27, 3)
Solution:
dy
=0
2x + 18y
dx
dy
−x
=
dx
9y
The tangent line has form y − 3 = −x
9y (x − 27), which is equivalent to
x 2 + 9y = 27x + 27y . Using the equation of the ellipse, 27x + 27y = 81.
So, 3 = x + y is one solution.
To find the other solution, solve for x in the ellipse and plug into the
equation for the tangent line. Use the points to find the tangent line slope.
MATH 125
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The University of Kansas
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