PRACTICE FINAL
INSTRUCTOR: PAUL APISA
Please write your name in the top right hand corner!
Problem 1 (7.5 points) : Use induction to show that
!
n
n
X
X
xi ≤
|sin xi |
sin
i=1
i=1
Problem 2 (10 points) : Write an -δ proof to prove that √
1
is continuous on
x+1
(−1, ∞).
Problem 3 (10 points): Prove, using the -δ definition of the limit, that if f is a
function that is differentiable at x0 and f 0 (x0 ) > 0 then there is a δ > 0 so that
f (x) < f (x0 ) for x in the interval (x0 − δ, x0 ).
Problem (10 points): Prove using the -δ definition of the limit that if f is continuous and lim g(x) = L, then
x−→c
lim f (g(x)) = f (L)
x−→c
Problem 4 (10 points)
: Evaluate these limits (2 points each)
√
arctan( x)
(1) lim
x−→0
x
sin2 (x)
(2) lim arcsin
x−→0
πx
sin(cos(x))
(3) lim
x−→0
x
sin(x)
(4) lim
x−→0 1 − cos(x)
x2 − 1
(5) lim
x−→1 x − 1
Problem 5 (5 points):
Using the limit definition of the derivative show that if f is differentiable and c is a
constant, then f (cx)0 = cf 0 (cx).
Problem 6 (5 points): Using the limit definition of the derivative show that f (x) =
x−1/2 is differentiable at all points in its domain.
Problem 7 (10 points) : Let a, b, and c be constants. Evaluate these derivatives
(2 points each).
(1) cota xb
(2) arctan sec2 (x) − 1
1
ax2 + bx + c
√
arccsc( x)
(4) arcsin (tan (xa ))
xa + sec(x)
(5) b
x + csc(x)
Problem 8 (5 points): Use the formula for differentiating the inverse of a function
to derive a formula for the derivative of arccsc(x).
Problem 9 (5 points): Suppose that f satisfies the equation
(3)
p(x) + q(f (x)) = r(x)f (x)
where all functions are differentiable. Find f 0 (x)
sin x
.
Problem 10 (10 points) : Let f (x) =
x
(1) (3 points) On what intervals is f strictly increasing?
(2) (3 points) On what intervals is f convex?
(3) (2 points) Find the critical points of f .
(4) (2 points) Using the second derivative test, determine whether each critical
point is a local maximum, a local minimum, or neither.
Problem 11 (5 points): Suppose that f is a convex function that is everywhere
positive and so that f (0) = f (1) = 1. Show that the area under the graph of f and
above the interval [0, 1] on the x-axis is less than one.
Problem 12 (7.5 points):
(1) (3.5 points) State the Mean Value Theorem (1 point) and use it to show that
| cos(x) − cos(y)| ≤ |x − y|
(2.5 points)
(2) (4 points) State the Cauchy Mean Value Theorem (1 point) and use it to show
that
x2
1−
≤ cos(x)
(3 points)
2
Bonus Problem (+3 points): Give an example of a convex function that is discontinuous.
Bonus Problem (+3 points): Suppose that n white dots and n black dots are
placed around a circle (with no two dots placed on top of each other). Prove
using induction that there is a point on the circle so that traveling clockwise around
the circle the number of black dots passed is always greater than or equal to the
number of white dots passed.
Write answers and work in the test booklet
2