1. Exercise 4: Part I
1.1. Implicit Differentiation. Let F (x, y) = 6x2 + 3xy + 2y 2 + 17y − 6. We call the set
C = {(x, y) : F (x, y) = 0}
the level curve of F. In this exercise, we are going to show that the level curve can be identified with
a function y = f (x) in a neighborhood of P (−1, 0).
(1) Show that F (−1, 0) = 0.
∂F
∂F
(2) Compute
and
∂x
∂y
∂F
(3) Show that
(P ) 6= 0.
∂y
(4) By (3) and using the implicit function theorem, we know that in a neighborhood of P,
the level curve defined a differentiable function y = f (x) so that 0 = f (−1). Compute
dy dx (x,y)=(−1,0)
(5) Find the tangent line and the normal line to C through P.
1.2. Mean Value Theorem.
(1) The polynomial
dn 2
(x − 1)n
dxn
is called the Legendre polynomial (of degree n.) Show that Pn (x) has n distinct zero in
(−1, 1).
(2) For x ≥ 2, prove that
π
π
(x + 1) cos
− x cos > 1.
x+1
x
(3) Let y = f (x) be a function continuous on [a, b] and differentiable on (a, b). Suppose that
f (a) = f (b) = 0. Show that there exists c ∈ (a, b) such that f 0 (c) = f (c).
Pn (x) =
1.3. Concavity. Let y = f (x) be a function defined on a closed interval [a, b] such that f 0 , f 00 exist
on (a, b) and f is continuous on [a, b]. We say that f is concave up if
f (tx1 + (1 − t)x2 ) ≤ tf (x1 ) + (1 − t)f (x2 ),
t ∈ [0, 1]
for any x1 , x2 ∈ [a, b] and concave down if
f (tx1 + (1 − t)x2 ) ≥ tf (x1 ) + (1 − t)f (x2 ),
t ∈ [0, 1]
00
for any x1 , x2 ∈ [a, b]. We have proved that if f > 0 on (a, b) then f is concave up on (a, b); if
f 00 < 0, then f is concave up on (a, b).
(1) Let f (x) = ex for x ∈ R. Show that f is concave up on R.
(2) Let f (x) = ln x, for x > 0. Show that f is concave down on x > 0.
(3) Suppose that a, b > 0. Use (2) to show that
√
a+b
ab ≤
.
2
(4) Suppose λ1 , λ2 , λ3 ≥ 0 such that λ1 + λ2 + λ3 = 1. Assume that f is concave up on (a, b).
Show that for any x1 , x2 , x3 ∈ (a, b), we have
f (λ1 x1 + λ2 x2 + λ3 x3 ) ≤ λ1 f (x1 ) + λ2 f (x2 ) + λ3 f (x3 ).
(5) Assume that f is concave up on (a, b). Show that for any λ1 , · · · , λn ≥ 0 with
and any x1 , · · · , xn ∈ (a, b), one has
!
n
n
X
X
f
λ i xi ≤
λi f (xi ).
i=1
i=1
This inequality is called the Jensen inequality.
1
Pn
i=1
λi = 1
2
(6) Assume that a1 , · · · , an > 0. Use (2) to show that
√
a1 + a2 + · · · + an
n
a1 a2 · · · an ≤
.
n
(7) If A, B, C are angles of a triangle, show that
√
3 3
sin A + sin B + sin C ≤
.
2
x4
− 2x2 + 4 for x ∈ R. Identify the intervals on which the function are concave
(a) Let f (x) =
4
up and concave down, decreasing and increasing. Also find all of its critical points, inflection
points.
1.4. Antiderivatives. Suppose that A = {F : [a, b] → R} and B = {G : [a, b] → R} are two sets of
functions. We define the sum A + B to be another set of functions
A + B = {(F + G) : [a, b] → R},
where (F + G)(x) = F (x) + G(x) for all x ∈ [a, b]. If k is a real number, we can also define kA to be
the set
(kA) = {(kF ) : [a, b] → R},
where (kF )(x) = kF (x) for x ∈ [a, b]. Let F, G be functions differentiable on (a, b) and continuous
on [a, b]. We know that (F + G)0 = F 0 + G0 and (aF )0 = a(F 0 ). From here, we know
Z
Z
Z
Z
Z
{f (x) + g(x)}dx = f (x)dx + g(x)dx,
af (x)dx = a f (x)dx.
(1) Compute the following indefinite integrals (Don’t use the method of change of variables,
integration by parts because we have not discussed it yet. Use the rules given above only.)
Z
Z
cos x
sin x
dx, I2 =
dx.
I1 =
sin x + cos x
sin x + cos x
(2) Do exercises in 4-1: 51, 55, 61, 62, 65