Mathematics

(52) d

dx

( 1

x2 · d

2

dx2

( 1

x2

)) = axp where a = and p = .

(53) Let f(x) = x+ 3

4− x . Find f (15)(x). Answer:

7n!

(4− x)p where n = and p = .

(54) Let f(x) = x

x+ 1 . Then f (4)(x) = a(x+ 1)p where a = and p = .

44 7. TECHNIQUES OF DIFFERENTIATION

(55) Let f(x) = x+ 1

2− x . Then f (4)(x) = a(2− x)p where a = and p = .

(56) Find the equation of the tangent line to the curve 2×6 + y4 = 9xy at the point (1, 2). Answer: 23y = ax+ b where a = and b = .

(57) For the curve x3 + 2xy + 13y 3 = 113 , find

dy

dx and

d2y

dx2 at the point (2,−1).

Answer: y′(2) = and y′′(2) = a

5 where a = .

(58) Find dy

dx and

d2y

dx2 for the devil’s curve y4 + 5y2 = x4 − 5×2 at the point (3, 2).

Answer: y′(3) = and y′′(3) = .

(59) Find dy

dx , d2y

dx2 , and

d3y

dx3 at the point (1, 8) on the astroid x2/3 + y2/3 = 5.

Answer: y′(1) = : y′′(1) = a

6 where a = ; and y′′′(1) =

b

24 where

b = .

(60) Find the point of intersection of the tangent lines to the curve x2 + y3− 3x+ 3y−xy = 18 at the points where the curve crosses the x-axis. Answer: ( , ).

(61) Find the equation of the tangent line to the curve x sin y + x3 = arctan ey + x− π4 at the point (1, 0). Answer: y = ax+ b where a = and b = .

(62) The equation of the tangent line to the lemniscate 3(x2 + y2)2 = 25(x2 − y2) at the point (2, 1) is y − 1 = m(x− 2) where m = .

(63) The points on the ovals of Cassini (x2+y2)2−4(x2−y2)+3 = 0 where there is a horizontal

tangent line are

( ± √ a

b √ b ,± 1

b √ b

) where a = and b = .

(64) The points on the ovals of Cassini (x2 + y2)2− 4(x2− y2) + 3 = 0 where there is a vertical tangent line are (±

√ a, b) and (±c, b) where a = , b = , and c = .

(65) At the point (1, 2) on the curve 4×2 + 2xy + y2 = 12, dy

dx = and

d2y

dx2 = .

(66) Let f and g be differentiable real valued functions on R. We know that the points (−4, 1) and (3, 4) lie on the graph of the curve y = f(x) and the points (−4, 3) and (3,−2) lie on the graph of y = g(x). We know also that f ′(−4) = 3, f ′(3) = −4, g′(−4) = −2, and g′(3) = 6. (a) If h = f · g , then h′(−4) = . (b) If j = (2f + 3g)4, then j ′(3) = .

(c) If k = f ◦ g , then k ′(−4) = .

(d) If ` = f

g , then ` ′(3) = .

(67) Let f(x) = 5 sinx+ 3 cosx. Then f (117)(π) = .

(68) Let f(x) = 4 cosx− 7 sinx. Then f (87)(0) = .

7.3. PROBLEMS 45

7.3. Problems

(1) Let (x0, y0) be a point in R2. How many tangent lines to the curve y = x2 pass through the point (x0, y0)? What are the equations of these lines? Hint. Consider the three cases: y0 > x0

2, y0 = x0 2, and y0 < x0

2.

(2) For the purposes of this problem you may assume that the differential equation

y′′ + y = 0 (∗)

has at least one nontrivial solution on the real line. (That is, there exists at least one twice differentiable function y, not identically zero, such that y′′(x) + y(x) = 0 for all x ∈ R.) (a) Show that if u and v are solutions of (∗) and a, b ∈ R, then w = au+ bv and u′ are

also solutions of (∗). (b) Show that if y is a solution of (∗) then y2 + (y′)2 is constant. (c) Show that if y is a nontrivial solution of (∗), then either y(0) 6= 0 or y′(0) 6= 0. Hint.

Argue by contradiction. Show that if y is a solution of (∗) such that both y(0) = 0 and y′(0) = 0, then y(x) = 0 for all x.

(d) Show that there exists a solution s of (∗) such that s(0) = 0 and s′(0) = 1. Hint. Let y be a nontrivial solution of (∗). Look for a solution s of the form a y + b y′ (with a, b ∈ R) satisfying the desired conditions.

(e) Show that if y is a solution of (∗) such that y(0) = a and y′(0) = b, then y = b s+a s′. Hint. Let u(x) = y(x)− b s(x)−a s′(x) and show that u is a solution of (∗) such that u(0) = u′(0) = 0. Use (c).

(f) Define c(x) = s′(x) for all x. Show that (s(x))2 + (c(x))2 = 1 for all x. (g) Show that s is an odd function and that c is even. Hint. To see that s is odd let

u(x) = s(−x) for all x. Show that u is a solution of (∗). Use (e). Once you know that s is odd, differentiate to see that c is even.

(h) Show that s(a + b) = s(a)c(b) + c(a)s(b) for all real numbers a and b. Hint. Let y(x) = s(x+ b) for all x. Show that y is a solution of (∗). Use (e).

(i) Show that c(a+b) = c(a)c(b)−s(a)s(b) for all real numbers a and b. Hint. Differentiate the formula for s(x+ b) that you derived in (h).

(j) Define t(x) = s(x)

c(x) and σ(x) =

1

c(x) for all x such that c(x) 6= 0. Show that

t′(x) = (σ(x))2 and σ′(x) = t(x)σ(x) wherever c(x) 6= 0. (k) Show that 1 + (t(x))2 = (σ(x))2 wherever c(x) 6= 0. (l) Explain carefully what the (mathematical) point of this problem is.

(3) Suppose that f is a differentiable function such that f ′(x) ≥ 3 2

for all x and that f(1) = 2.

Prove that f(5) ≥ 8. (4) Suppose that f is a differentiable function such that f ′(x) ≥ 3 for all x and that f(0) = −4.

Prove that f(3) ≥ 5. (5) Suppose that f is a differentiable function such that f ′(x) ≤ −2 for all x ∈ [0, 4] and that

f(1) = 6. (a) Prove that f(4) ≤ 0. (b) Prove that f(0) ≥ 8.

(6) Give a careful proof that sinx ≤ x for all x ≥ 0. (7) Give a careful proof that 1− cosx ≤ x for all x ≥ 0.

(8) Prove that if x2 = 1− y2

1 + y2 , then

( dx

dy

)2 =

1− x4

1− y4 at points where y 6= ±1.

46 7. TECHNIQUES OF DIFFERENTIATION

(9) For the circle x2 + y2 − 1 = 0 use implicit differentiation to show that y′′ = − 1 y3

and

y′′′ = −3x y5

.

(10) Explain how to calculate d2y

dx2 at the point on the folium of Déscartes

x3 + y3 = 9xy

where the tangent line is parallel to the asymptote of the folium.

(11) Explain carefully how to find the curve passing through the point (2, 3) which has the following property: the segment of any tangent line to the curve contained between the (positive) coordinate axes is bisected at the point of tangency. Carry out the computation you have described.

7.4. ANSWERS TO ODD-NUMBERED EXERCISES 47

7.4. Answers to Odd-Numbered Exercises

(1) −5 2

, −3 2

, 9, 1

2

(3) 4, −2 3

, −6, −8 5

(5) 36

(7) 13

(9) −3, 0, 2 (11) −1, 2 (13) 4, 12

(15) 0, 0, −2 (17) −1, 1, −1, 1 (19) 9, 3

(21) 2×5 + 14×3 − 10x, 4×3 + 14x

(23) sin3 x, tanx, sec2 x, ( x3 − 3×2 + 6x− 11

) 2 3 , x2 − 2x+ 2, −1

2

(25) 100(2− π) (27) 200, 401

(29) 12, −35 (31) −3, 0, −1, 1 (first row)

2, 2, 1, 1 (second row) 1, 3 (third row) 13, 8

(33) 3 ln 3

(35) 1, ∞, ln 2, 2e2

(37) 1, ln sinx

(39) 1, 9, 18, 6

(41) 8

(43) 2, 2

3

(45) 1, −3, −2 (47) 11, 10

(49) −1, 2 (51) −2, 4 (53) 15, 16

(55) 72, −5 (57) −2, 4 (59) −2, 5, −25 (61) −4, 4 (63) 15, 2

48 7. TECHNIQUES OF DIFFERENTIATION

(65) −2, −4 3

(67) −5

CHAPTER 8

THE MEAN VALUE THEOREM

8.1. Background

Topics: Rolle’s theorem, the mean value theorem, the intermediate value theorem.

8.1.1. Definition. A real valued function f defined on an interval J is increasing on J if f(a) ≤ f(b) whenever a, b ∈ J and a ≤ b. It is strictly increasing on J if f(a) < f(b) whenever a, b ∈ J and a < b. The function f is decreasing on J if f(a) ≥ f(b) whenever a, b ∈ J and a ≤ b. It is strictly decreasing on J if f(a) > f(b) whenever a, b ∈ J and a < b. NOTE: In many texts the word “nondecreasing” is used where “increasing” in these notes; and “increasing” is used for “strictly increasing”.

49

50 8. THE MEAN VALUE THEOREM

8.2. Exercises

(1) Let M > 0 and f(x) = x3 for 0 ≤ x ≤M . Find a value of c which satisfies the conclusion of the mean value theorem for the function f over the interval [0,M ]. Answer: c =

M

a where a = .

(2) Let f(x) = x4 + x+ 3 for 0 ≤ x ≤ 2. Find a point c whose existence is guaranteed by the mean value theorem. Answer: c = 2p where p = .

(3) Let f(x) = √ x for 4 ≤ x ≤ 16. Find a point c whose existence is guaranteed by the mean

value theorem. Answer: c = .

(4) Let f(x) = x

x+ 1 for −12 ≤ x ≤

1 2 . Find a point c whose existence is guaranteed by the

mean value theorem. Answer: c = a

2 − 1 where a = .

8.3. PROBLEMS 51

8.3. Problems

(1) Use Rolle’s theorem to derive the mean value theorem.

(2) Use the mean value theorem to derive Rolle’s theorem.

(3) Use the mean value theorem to prove that if a function f has a positive derivative at every point in an interval, then it is increasing on that interval.

(4) Let a ∈ R. Prove that if f and g are differentiable functions with f ′(x) ≤ g′(x) for every x in some interval containing a and if f(a) = g(a), then f(x) ≤ g(x) for every x in the interval such that x ≥ a.

(5) Suppose that f is a differentiable function such that f ′(x) ≤ −2 for all x ∈ [0, 4] and that f(1) = 6. (a) Prove that f(4) ≤ 0. (b) Prove that f(0) ≥ 8.

(6) Your friend Fred is confused. The function f : x 7→ x 2 3 takes on the same values at x = −1

and at x = 1. So, he concludes, according to Rolle’s theorem there should be a point c in the open interval (−1, 1) where f ′(c) = 0. But he cannot find such a point. Help your friend out.

(7) Consider the equation cosx = 2x. (a) Use the intermediate value theorem to show that the equation has at least one solution. (b) Use the mean value theorem to show that the equation has at most one solution.

(8) Let m ∈ R. Use Rolle’s theorem to show that the function f defined by f(x) = x3−3x+m can not have two zeros in the interval [−1, 1].

(9) Use the mean value theorem to show that if 0 < x ≤ π/3, then 12x ≤ sinx ≤ x. (10) Use the mean value theorem to show that on the interval [0, π/4] the graph of the curve

y = tanx lies between the lines y = x and y = 2x.

(11) Let x > 0. Use the mean value theorem to show that x

x2 + 1 < arctanx < x.

(12) Use the mean value theorem to show that

x+ 1 < ex < 2x+ 1

whenever 0 < x ≤ ln 2. (13) Show that the equation ex + x = 0 has exactly one solution. Locate this solution between

consecutive integers.

(14) Prove that the equation sinx = 1 − 2x has exactly one solution. Explain how the inter- mediate value theorem can be used to produce an approximation to the solution which is correct to two decimal places.

(15) Give a careful proof that at one time your height (in inches) was exactly equal to your weight (in pounds). Be explicit about any physical assumptions you make.

52 8. THE MEAN VALUE THEOREM

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