1. Consider the function f(x) = x
3 + 7x + 2.
(a) (10 pts) Show that f has an inverse. (Hint: Is f increasing?)
(b) (10 pts) Determine the slope of the tangent line to f
−1 at (10, 1). (Hint: The
derivative of the inverse function can be found using the chain rule.)
2. (10 pts) Determine the equation of the tangent line to the curve 2xy − 3y
2 = 9 at
A(6, 1).
3. (10 pts) Write down the quadratic approximation to f(x) = lnx + e
x around x = 1.
4. (10 pts) Show that the equation 3x − x
5 + 2x + 2 = 0 has at least one solution between
0 and 2. (Remember the Intermediate Value Theorem)
5. (10 pts) Evaluate the following limit: limx→0
e
x − e
−x
x
.
6. (10 pts) Consider the function f(x) = x
3+x−2x
1
3 +4. Show that there exists x between
0 and 1 such that the tangent line drawn to f at x is horizontal. (Hint: Remember the
Mean Value Theorem)
7. Consider the function given by: f(x) = 4x +
1
x
defined on the interval (0, ∞).
(a) (10 pts) Determine the concavity/convexity of f.
(b) (10 pts) Find the all extreme points of f and determine whether each is a maximum
or a minimum point.
8. (10 pts) Find all inflection points of the function f(x) = x
4 − 4x
3 − 18x
2 + 6
İşin Yapılacağı Konum: İSTANBUL
Görevin Başlangıç Tarihi: 09-07-2020
Görevin Bitiş Tarihi: 09-07-2020
Kategori: Ders / Çeviri