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Calculus homework help. GET INSPIRED! PLEASE DO NOT COPY.
Name:
Honor Code: On my honor, I neither received nor given any aid during this examination.
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INSTRUCTIONS
• PLEASE READ THE INSTRUCTIONS BELOW.
• All communication devices (i.e, cell phones, Apple watches, etc . . .) have to be turned off,
and stowed away from your desk.
• You are not permitted to leave the classroom until you have turned in your exam to the
instructor.
• No calculators allowed.
• Show all work to get full credit.
• All answers have to be simplified, and in exact form, i.e, one fraction with positive exponents, and in lowest form when applicable.
GOOD LUCK!
1
Problem 1. (20pt) Solve the following differential equations:
(1 + x
2
)
2
dy
dx + 2x + 2xy2 = 0
p
1 + x
2
dy
dx − xy3 = 0
Problem 2. (20pt) Use Taylor series to calculate the following limitation, choose one of questions
to do according to your exam version.
(A) lim
x→0
e
x − 1 − x

1 − x − cos √
x
(B) lim
x→0
x
2
2 + 1 −

1 + x
2
(cos x − e
x2
) sin x
2
Problem 3. (20pt) Calculate the Taylor formula for the following functions at x0 until the given
derivative order n, choose one of questions to do according to your exam version.
• (A) 1 + 2x − 3x
2 + 5x
3 at x0 = 1
• (B) e
sin x at x0 = 0 until n = 4
2
Problem 4. (20pt) Calculate the arc length of the following curves.
• y = x
3
2 for 2 6 x 6 3
• x(t) = t + e
−t and y(t) = t − e
−t
for 0 6 t 6 2
Problem 5. (20pt) Use the following ideas to prove:
Assume f(x) is a smooth function defined on R,
lim
h→0
f(x + h) − 2f(x) + f(x − h)
h
2
= f
00(x)
for any x ∈ R.
Here is the rough ideas: By the Taylor formula, for any x0, we have
f(x) = Xn
k=0
f
(k)
(x0)
k!
(x − x0)
k + Rn(x)
now let x − x0 = h, thus x = x0 + h, we can get
f(x0 + h) = Xn
k=0
f
(k)
(x0)
k!
h
k + Rn(x0 + h)
let x = x0, we get
f(x + h) = Xn
k=0
f
(k)
(x)
k!
h
k + Rn(x + h)
and
lim
h→0
Rn(x + h)
h
n
= 0
now you can use the results above to get the taylor formula for f(x − h) and expand both f(x + h)
and f(x − h) until derivative order 2, then prove the equality above.

Calculus homework help

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