1. Exercise 4: Part I
1.1. Implicit Differentiation. Let F (x, y) = 6x2+ 3xy + 2y2+ 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.
(2) Compute ∂F
∂x and ∂F
∂y (3) Show that ∂F
∂y(P ) 6= 0.
(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
Pn(x) = dn
dxn(x2− 1)n
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
(x + 1) cos π
x + 1 − x cosπ x > 1.
(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 f0(c) = f (c).
1.3. Concavity. Let y = f (x) be a function defined on a closed interval [a, b] such that f0, f00 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]
for any x1, x2 ∈ [a, b]. We have proved that if f00 > 0 on (a, b) then f is concave up on (a, b); if f00< 0, then f is concave up on (a, b).
(1) Let f (x) = exfor 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
√
ab ≤ a + b 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 (λ1x1+ λ2x2+ λ3x3) ≤ λ1f (x1) + λ2f (x2) + λ3f (x3).
(5) Assume that f is concave up on (a, b). Show that for any λ1, · · · , λn ≥ 0 withPn
i=1λi = 1 and any x1, · · · , xn∈ (a, b), one has
f
n
X
i=1
λixi
!
≤
n
X
i=1
λif (xi).
This inequality is called the Jensen inequality.
1
2
(6) Assume that a1, · · · , an> 0. Use (2) to show that
√n
a1a2· · · an≤a1+ a2+ · · · + an
n .
(7) If A, B, C are angles of a triangle, show that
sin A + sin B + sin C ≤ 3√ 3 2 . (a) Let f (x) = x4
4 − 2x2+ 4 for x ∈ R. Identify the intervals on which the function are concave 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= F0+ G0 and (aF )0= a(F0). From here, we know
Z
{f (x) + g(x)}dx = Z
f (x)dx + Z
g(x)dx, Z
af (x)dx = a Z
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.)
I1=
Z sin x
sin x + cos xdx, I2=
Z cos x
sin x + cos xdx.
(2) Do exercises in 4-1: 51, 55, 61, 62, 65
