Written Homework 11
Math 1200
Instructions: Submit your work on Gradescope bySunday, November 3, 2024 at 11:59
pm. You must show all work. Due to the upcoming exam, solutions have already been
posted, and the assignment will be graded entirely on completion.
1.Consider the curvey
2
−5x=xy−15. Find the equation of the tangent line of the
curvey
2
−5x=xy−15 at all points of the form (4, y).
1
2. Consider the curvey
2
+ 3x=x
3
+ 3.
(a) Find all points on the curve of the form (0, y).
(b)
Find the equation of
dy
dx
, and use it to find the slope of the tangent line of the
curve at the points you found in the previous part.
(c) Find all points (x, y) where the tangent line of the curve is horizontal.
2
3.The following table gives some values of an invertible functionf(x) and its derivative
f
′
(x).
x
35810
f(x)58103
f
′
(x)−235−1
Compute the following derivatives.
(a)g
′
(8),whereg(x) = ln(f(x)).
(b)j
′
(3),wherej(x) = 5
x
log
3
(f(x))
(c) (f
−1
)
′
(8)
4. Compute the derivative of each of the functiony=x
e
x
3
5. Order the following functions from least dominant to most dominant:
(a)f(x) = log
3
(x) + 5
(b)g(x) =x
35
+x
5
+ 12
(c)h(x) = ln(x) + 35x
3
+ 8
(d)j(x) = 2
x
−x
27
+ 5(4
x
)
(e)k(x) = 5
x
+ 10
(f)m(x) = log
4
(5)·x−8
6. Compute
dy
dx
for the curve defined byx
2
lny+ 1 = ln(xy
2
).
4
7. Compute the following limits:
(a) lim
x→0
ln(x
2
+ 1)
x
2
−2x
(b) lim
x→∞
ln(2x+ 5e
x
)
ln(4e
2x
+ 7x)
5
