MA113 Test 4 Solns - University of Kentucky

MA 113 -- Calculus I Exam 4

Fall 2009 December 15, 2009

Answer all of the questions 1 - 7 and two of the questions 8 - 10. Please indicate which problem is not to be graded by crossing through its number in the table below.

Additional sheets are available if necessary. No books or notes may be used. Please, turn off your cell phones and do not wear ear-plugs during the exam. You may use a calculator, but not one which has symbolic manipulation capabilities. Please:

(a) clearly indicate your answer and the reasoning used to arrive at that answer (unsupported answers may not receive credit),

(b) give exact answers, rather than decimal approximations to the answer(unless otherwise stated).

Each question is followed by space to write your answer. Please write your solutions neatly in the space below the question. You are not expected to write your solution next to the statement of the question.

Name:

Key

Section:

Last four digits of student identification number: floor[frac[e]x104]

Question

1 2 3 4 5 6 7 8 9 10 Free

Score 3

Total

9 13 8 6 12 9 10 15 15 15 3

100

1

1. Find the following limits:

(a)

lim

x0

2

sin

x

sin(2 5x

x

)

ln 2 e2x

(b) lim x

5x

(c)

lim e2x 1 x0 tan x

(a)

lim

x0

2sinx

- sin(2x) 5x

H

lim

x0

2cosx

-

2cos(2x) 5

=0

2

:

1 1

: :

both derivatives answer

correct

(b)

lim ln

x

2 + e2x 5x

2e2x

H

lim

x

2

+ e2x 5

=

lim

x

2e2x 10 + 5e2x

H

lim

x

4e2x 10e2x

=

2 5

1 : both derivatives correct (1st application)

4

:

1 1

: :

a lg ebra both derivatives

correct

(2nd

application)

1 : answer

(c)

lim

x0

e2x 1 tan x

H

lim

x0

2e2x sec2 x

2

2

:

1 1

: :

both derivatives answer

correct

1: mention at some point the applicability of l'Hospital's Rule

(a)

lim

x 0

2

sin

x

sin(2x 5x

)

ln 2 e2x

(b) lim x

5x

(c)

lim

x0

e2x 1 tan x

0 2/5 2

2

Total: 9

2. Consider the function g(x) x3 6x2 9x 2 on the closed interval [?1,4].

(a) List all of the critical points of g(x) . (b) On what intervals is g(x) increasing? (c) On what intervals is g(x) concave down?

(d) List all x- and y-coordinates for the absolute maximum. (e) List all x- and y-coordinates for the absolute minimum.

(a)

g(x) = 3x2 12x 9

g(x) 0

3(x 1)(x 3) 0

x 1,3

2

:

1 1

: :

g(x) both

correct critical

points

(b) The function is increasing on the intervals ?1 < x < 1 and 3 < x < 4 because the g' > 0 there.

3

:

2 : (-1,1) 1 : reason

(3,

4)

(c) g(x) = 6x 12 , so g(x) is concave down

on (?1,2), since g'' < 0 there.

2

:

1 1

: :

2nd derivative correct answer and reason

(d) Evaluate g(?1) = ?14, g(1) = 6 and g(4) = 6. The absolute maximum occurs at (1,6) and at (4,6).

4

:

2 2

: :

(1,6) (4,6)

(e) Evaluate g(?1) = ?14, g(3) = 2 and g(4) = 6. The absolute minimum occurs at (?1,?14).

2: (?1,?14)

(a) Critical points: x = 1,3

(b) g(x) is increasing on (?1,1) U (3,4)

(c) g(x) is concave down on

(?1,2)

(d) The absolute maximum occurs at the point(s): (1,6) and (4,6)

(e) The absolute minimum occurs at the point(s): (?1,?14)

3

Total: 13

(c) Consider the curve y2 xey 1 .

(a)

Find the derivative,

dy dx

,

of y.

(b) Find the slope of the tangent line to this curve at the point (1,0).

(c) Find the equation of the tangent line at the point (1,0). Express it in the form y = mx + b.

(a)

y2 xey = 1

2y

dy dx

ey

xey

dy dx

0

dy

ey

dx 2y xey

1 : use implicit differentiation

4

:

1 1

: :

d(y2) 2yy product rule

and

d(xey )

1 : answer

(b)

m

dy dx x1,y0

e0 2(0) 1?e0

1

(c) y 0 = -1(x 1) y x 1

2

:

1 1

: :

evaluate answer

derivative

at

(1, 0)

2

:

1 1

: :

use point answer

(1,0)

and

slope

-1

(a)

dy

ey

dx 2y xey

(b) The slope of the tangent line: m=

(c) The equation of the tangent line:

?1 y = ?x + 1

4

Total: 8

4.

(a) If f (x) x10g(x) , g(1) 2 , and g(1) 3 , find f (1) .

(b) If h(x) cos x2 ex2 , find h(x) .

(a) f(x) = x10g(x) f(x) 10x9g(x) x10g(x) f'(1) 10 (1)9 g(1) 110 g(1) 20 3 23

(b) h(x) cos x2 ex2

h'(x) sin(x2 ex2 )(2x 2xex2 )

1 : correct derivative 3 : 1 : substitute x 1

1 : answer 1 : derivative of outside function 3 : 1 : derivative of inside function 1 : answer

(a) f (1) 23 (b) h(x) sin(x2 ex2 )(2x 2xex2 )

5

Total: 6

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