Math 1111 Calculus (I)
Homework 14 1. (a) Letf(x) =
Z 3x
2x
u2−1
u2+ 1 du. Findf0(x).
(b) Letf =
Z 2 sinx
cos2(3x−1)
ln(20 +t3) dt. Find f0(x).
(c) Determine the interval where the curve y= Z x
0
t2
t2+t+ 2 dt is concave downward.
(d) Iff(x) =Rsinx 0
√1 +t2 dt and g(y) =Ry
3 f(x) dx, find g00(π 6).
(e) Let f(x) = Z x2
1
√
t3+ 3t+ 12 dt. Prove that f is invertible on (0,∞) and find the equation of the tangent line of y = f−1(x) passing (0,1).
2. Find the general indefinite integral R
f(x) dx for givenf. (a) f(x) = (x+ 4)(2x+ 1)
(b) f(x) = x3−2√ x x
(c) f(x) = secx(secx+ tanx) + 2 x2+ 1 3. Find the definite integralRb
a f(x) dx for givenf on [a, b].
(a) f(x) = x √3 x+√4
x
on [0,1]
(b) f(x) = |2x−1| on [0,2].
(c) f(x) = |sinx| on [0,3π 2 ].
(d) f(x) = sinx+ sinxtan2x
sec2x on [0,π 3].
4. Letf be a function defined on [a, b]. Prove that any one of the following statements implies that f is bounded on [a, b].
(Note: you can only assume thatf has one of the following conditions.) (a) f is continuous on [a, b].
(b) f is increasing on [a, b].
(c) f is uniformly continuous on (a, b).
(d) f is differentiable on (a, b) and |f0(x)|< M for all x∈(a, b).
5. Define f(x) = 1
x2 on (−∞,0)∪(0,∞).
(a) Find an antiderivativeg of the function f.
(b) Based on the antiderivativeg you find in problem (a), check that g1(x) :=
g(x), x >0 g(x) + 1, x <0 is also an antiderivative off.
(c) The second Fundamental Theorem of Calculus suggests that Z 2
−2
f(x) dx=g(2)−g(−2) and
Z 2
−2
f(x)dx=g1(2)−g1(−2).
Check your answers.
(d) If the above answers are not equal, explain the reason why the second Fundamental Theorem of Calculus fails.
6. (a) Prove that lim
x→0+lnx=−∞.
(b) Prove that lnx < xfor all x >0.
(Hint: when 0 < x ≤ 1, it is clear; when x > 1, observe that the area of the corresponding region is less then x−1. )
(c) Forn ∈N, prove that lnx < x1n for all sufficiently large x.
(Hint: whenn = 1,2, it is easy; when n≥3, let f(x) =xn1 −lnx and show that f(nn2)>0 and f0(x)>0 for x≥nn2.)
(d) Without using L’Hˆopital’s Rule to prove
x→∞lim lnx
xn1 = 0 for any n ∈N. (Hint: Express lnx
x1/n by lnx x1/2n · 1
x1/2n and consider the result of problem (c).)
7. Prove that following equalities hold. (Hint: Differentiate both sides and use some results of differentiation. )
(a) Z x
0
cos3t dt= sinx− 1 3sin3x (b)
Z x
0
tant dt=−ln|cosx| for x∈(−π 2,π
2).
(c) Letfbe differentiable on (0,∞) withf(xy) = f(x)+f(y) · · ·(∗) for all x, y >0. Prove that
f(x) =Clnx for some constant C. (1) (Hint: (i)f(1) =f(1·1) = f(1)+f(1) ⇒f(1) = 0. (ii) Differen- tiate ( d
dx) both sides of (∗)and then takex= 1 to provef0(y) = C y for any yand some constantC. (iii) Letg(x) =f(x)−Clnxand consider g0(x).)
(d) Let f be differentiable on R with f(x+y) =f(x)f(y) · · ·(∗∗) for all x, y ∈R. Prove that
f(x)≡0 or f(x) = exp(Cx) for some constant C. (2) (Hint: (i) f(0) = f(0 + 0) = f(0)f(0) ⇒ f(0) = 0 or 1. (ii) When f(0) = 0, show that f(x) ≡ 0. (iii) When f(0) = 1, show that f(x) 6= 0 and f(x) > 0 for all x. (iv) Use the definition of differentiation and(∗∗)to show that f0(x) =Cf(x) for some con- stant C. (v) By using the result d
dx
lnf(x)
= f0(x)
f(x) to complete this problem.)
Remark: For Problem7(c), it is easy to check that if (1) holds, then the hypothesis of Problem7(c) is true. Hence, we obtain
“⇔ ” for Problem7(c). The similar consequence holds for Prob- lem7(d).
