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Calculus: Identifying Local and Absolute Extrema and Intervals of Increase and Decrease - , Assignments of Analytical Geometry and Calculus

University of Wisconsin (UW) - Madison Analytical Geometry and Calculus

Prof. Bokai Yan

Solutions to various calculus problems related to identifying local and absolute extrema, intervals of increase and decrease, and applying the mean value theorem. Topics include finding maximum and minimum values, using the first derivative test, and applying rolle's theorem.

Typology: Assignments

Pre 2010

Uploaded on 09/02/2009

koofers-user-ar1 🇺🇸

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Sec. 4.1

3. absolute maximum at x=c; no absolute minimum. The interval is not closed.

4. no absolute maximum or absolute minimum. The interval is not closed and h(x) is not

continuous.

11. (c) 12. (b) 13. (d) 14. (a)

18. absolute maximum = 4, absolute minimum = 5

−

;

19. absolute maximum = , absolute minimum =

1/4−4

−

;

24. absolute maximum = 0, absolute minimum = 5

−

;

27. absolute maximum = 2/ 3, absolute minimum = 1;

Discover Assignments of Analytical Geometry and Calculus University of Wisconsin (UW) - Madison

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Sec. 4.

absolute maximum at x=c; no absolute minimum. The interval is not closed.
no absolute maximum or absolute minimum. The interval is not closed and h(x) is not continuous.
(c) 12. (b) 13. (d) 14. (a)
absolute maximum = 4, absolute minimum = −^5 ;
absolute maximum = −1/ 4^ , absolute minimum = −^4 ;
absolute maximum = 0, absolute minimum = − 5 ;
absolute maximum = 2 / 3 , absolute minimum = 1;

absolute maximum = 2, absolute minimum = 0/
absolute maximum = 1 at θ = 1 , absolute minimum = − 8 at θ = − 32.
local maximum = 4 6 4 9

at 6 3

x =

local minimum = 4 6 4 9

− + at 6 3

x = −

local minimum = 1 at x = 0.
20 7

mi along the coast from A to B.

62. (a) A = 2 r ( 200 − π r ), 0 r^200

or A^2 x ( 200 x )

= − , 0 ≤^ x ≤^200

(b) When r^100 m

= , x = 100 m , maximum A^2000 m^2

Sec. 4.

c = ⎛⎜^ ⎞⎟ ⎝ ⎠

a^ =^3 , b^ =^4 , m^ =^1
(a) See the answer in textbook. (b) Prove:

Let P x ( ) = x n^ + an − 1 x n −^1 + "+ a x 1 + a 0

Then P ′( x ) = nxn −^1 + ( n − 1 ) an − 1 xn −^2 + "+ a 1

Suppose x 1 , x 2 are any two zeros of P x (^ ), i.e. P x ( 1^ ) =^ P x ( 2 )=^0

By Rolle’s Theorem, there exists y between x 1 and x 2 , such that

P ′^ ( y ) = 0.

the speed limit 65 mph.

(a) increasing: x ∈ \

(b) There are no local extrema.

(a) increasing: x ∈ \

Sec. 4.
1. (a) − 2 ,
- (b) increasing: x >^1 and x < −
  - decreasing: − 2 < x <
- (c) local minimum at x =
  - local maximum at x = -
1. (a) − 2 ,
- (b) increasing: x > −
  - decreasing: x < −
- (c) local minimum at x = -
1. (a) − 5 , − 1 ,
- (b) increasing: − 5 < x < − 1 and x >
  - decreasing: x < −^5 and −^1 < x <
- (c) local minimum at x = -5,
  - local maximum at x = -
1. (a) increasing: x < − 3 and x >
  - decreasing: − 3 < x <
- (b) local minimum: h ( 3 )= −
  - local maximum: h ( − 3 )=
1. (a) increasing: −^2 < x < (c) none
  - decreasing: − 2 2 < x < − 2 and 2 < x <
- (b) local minimum: g ( − 2 )= − 4 , g ( 2 2 )=
  - local maximum: g ( 2 )= 4 , g ( − 2 2 )=
- (c) absolute minimum: g ( −^2 ) = −
  - absolute maximum: g ( 2 ) =