冉 冊 冉 冊 DIAGNOSTIC TESTS

A DIAGNOSTIC TEST: ALGEBRA

1. Evaluate each expression without using a calculator.

(a) (b) (c)

(d) (e) (f )

2. Simplify each expression. Write your answer without negative exponents.

(a) (b) (c)

3. Expand and simplify.

(a) (b)

(c) (d)

(e)

4. Factor each expression.

(a) (b)

(c) (d)

(e) (f )

5. Simplify the rational expression.

(a) (b)

(c) (d)

6. Rationalize the expression and simplify.

(a) (b)

7. Rewrite by completing the square.

(a) (b)

共3兲⁴ 3⁴ 3^4

5²³

5²¹

冉

冊

s200  s32 共3a³b³兲共4ab²兲²

冉

冊

3共x  6兲  4共2x  5兲 共x  3兲共4x  5兲

(

)(

)

共x  2兲³

4x² 25 2x² 5x  12

x³ 3x² 4x  12 x⁴ 27x 3x^3兾2 9x^1兾2 6x^1兾2 x³y 4xy

x² 3x  2 x² x  2

2x² x  1

x² 9 ⴢ x 3 2x 1

x²

x² 4  x 1 x 2

y x  x

y 1 y  1

x

s4 h  2 h s10

s5  2

2x² 12x  11 x² x  1

DIAGNOSTIC TESTS

Success in calculus depends to a large extent on knowledge of the mathematics that precedes calculus:

algebra, analytic geometry, functions, and trigonometry. The following tests are intended to diagnose weaknesses that you might have in these areas. After taking each test you can check your answers against the given answers and, if necessary, refresh your skills by referring to the review materials that are provided.

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8. Solve the equation. (Find only the real solutions.)

(a) (b)

(c) (d)

(e) (f )

(g)

9. Solve each inequality. Write your answer using interval notation.

(a) (b)

(c) (d)

(e)

10. State whether each equation is true or false.

(a) (b)

(c) (d)

(e) (f )

x 5 苷 14 ¹2x 2x

x 1 苷 2x 1 x

x² x  12 苷 0 2x² 4x  1 苷 0

x⁴ 3x² 2 苷 0 3

ⱍ

ⱍ

2x共4  x兲^1兾2 3s4  x 苷 0

4  5  3x  17 x² 2x  8

x共x  1兲共x  2兲  0

ⱍ

ⱍ

2x 3 x 1  1

共p  q兲²苷 p² q² sab 苷 sa sb

sa² b² 苷 a  b 1 TC

C 苷 1  T 1

x y 苷 1 x  1

y

1兾x

a兾x  b兾x 苷 1 a b xviii DIAGNOSTIC TESTS

A ANSWERS TO DIAGNOSTIC TEST A: ALGEBRA

1. (a) (b) (c)

(d) (e) (f )

2. (a) (b) (c)

3. (a) (b)

(c) (d)

(e)

4. (a) (b)

(c) (d)

(e) (f )

5. (a) (b)

(c) (d)

81 81 ⁸¹¹

25 ⁹4

1 8

6s2 48a⁵b⁷ x

9y⁷ 11x 2 4x² 7x  15 a b 4x² 12x  9 x³ 6x² 12x  8

共2x  5兲共2x  5兲 共2x  3兲共x  4兲 共x  3兲共x  2兲共x  2兲 x共x  3兲共x² 3x  9兲 3x^1兾2共x  1兲共x  2兲 xy共x  2兲共x  2兲

x 2 x 2

x 1 x 3 1

x 2 共x  y兲

6. (a) (b)

7. (a) (b)

8. (a) (b) (c)

(d) (e) (f )

(g)

9. (a) (b)

(c) (d)

(e)

10. (a) False (b) True (c) False

(d) False (e) False (f ) True

5s2  2s10 1

s4 h  2

(

)

6 1 3, 4

1 ¹2s2 1, s2 ²3, ²²₃

12 5

关4, 3兲 共2, 4兲

共2, 0兲 傼 共1, 兲 共1, 7兲 共1, 4兴

If you have had difficulty with these problems, you may wish to consult the Review of Algebra on the website www.stewartcalculus.com.

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B DIAGNOSTIC TEST: ANALYTIC GEOMETRY

1. Find an equation for the line that passes through the point and (a) has slope

(b) is parallel to the -axis (c) is parallel to the -axis (d) is parallel to the line

2. Find an equation for the circle that has center and passes through the point .

3. Find the center and radius of the circle with equation . 4. Let and be points in the plane.

(a) Find the slope of the line that contains and .

(b) Find an equation of the line that passes through and . What are the intercepts?

(c) Find the midpoint of the segment . (d) Find the length of the segment .

(e) Find an equation of the perpendicular bisector of . (f ) Find an equation of the circle for which is a diameter.

5. Sketch the region in the -plane defined by the equation or inequalities.

(a) (b)

(c) (d)

(e) (f )

共2, 5兲

3 x y

2x 4y 苷 3

共1, 4兲 共3, 2兲

x² y² 6x  10y  9 苷 0 A共7, 4兲 B共5, 12兲

A B

A B

AB AB

AB AB xy

1  y  3

ⱍ

ⱍ

ⱍ

ⱍ

y 1 ¹2x y x² 1

x² y² 4 9x² 16y²苷 144

DIAGNOSTIC TESTS xix

B ANSWERS TO DIAGNOSTIC TEST B: ANALYTIC GEOMETRY

1. (a) (b)

(c) (d)

2.

3. Center , radius 5 4. (a)

(b) ; -intercept , -intercept

(c) (d) (e) (f )

y苷 3x  1 y苷 5

x苷 2 y苷¹2x 6

共x  1兲² 共y  4兲²苷 52 共3, 5兲

⁴3

4x 3y  16 苷 0 x 4 y ¹⁶3

共1, 4兲 20

3x 4y 苷 13

共x  1兲² 共y  4兲²苷 100

5.

y

x

1 2

0

y

0 x

y

0 4 x

3

_1

2 y

x 0

y

0 4x

_4

y

0 2 x 1

(a) (b) (c)

(d) (e) (f )

_1 3

2

_2

y=≈-1

≈+¥=4

y=1- x¹₂

If you have had difficulty with these problems, you may wish to consult the review of analytic geometry on the website www.stewartcalculus.com.

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C DIAGNOSTIC TEST: FUNCTIONS

1. The graph of a function is given at the left.

(a) State the value of . (b) Estimate the value of . (c) For what values of is ?

(d) Estimate the values of such that . (e) State the domain and range of .

2. If , evaluate the difference quotient and simplify your answer.

3. Find the domain of the function.

(a) (b) (c)

4. How are graphs of the functions obtained from the graph of ?

(a) (b) (c)

5. Without using a calculator, make a rough sketch of the graph.

(a) (b) (c)

(d) (e) (f )

(g) (h)

6. Let

(a) Evaluate and . (b) Sketch the graph of .

7. If and , find each of the following functions.

(a) (b) (c)

f f共1兲

f共2兲 x f共x兲 苷 2

x f共x兲 苷 0

f

f共x兲 苷 x³ f共2  h兲  f 共2兲

h

f共x兲 苷 2x 1

x² x  2 t共x兲 苷 s³x

x² 1 h共x兲 苷 s4  x  sx² 1 f

y苷 f 共x兲 y苷 2 f 共x兲  1 y苷 f 共x  3兲  2

y苷 x³ y苷 共x  1兲³ y苷 共x  2兲³ 3

y苷 4  x² y苷 sx y苷 2sx

y苷 2^x y苷 1  x^1

f共x兲 苷

再

f共2兲 f共1兲 f

f共x兲 苷 x² 2x  1 t共x兲 苷 2x  3

fⴰ t tⴰ f tⴰ t ⴰ t

xx DIAGNOSTIC TESTS

y

0 x

1 1

FIGURE FOR PROBLEM 1

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DIAGNOSTIC TESTS xxi

C ANSWERS TO DIAGNOSTIC TEST C: FUNCTIONS

1. (a) (b) 2.8

(c) (d)

(e) 2.

3. (a) (b) (c)

4. (a) Reflect about the -axis

(b) Stretch vertically by a factor of 2, then shift 1 unit downward

(c) Shift 3 units to the right and 2 units upward 5.

2

3, 1 2.5, 0.3

关3, 3兴, 关2, 3兴 12 6h  h²

共, 2兲 傼 共2, 1兲 傼 共1, 兲 共, 兲

共, 1兴 傼 关1, 4兴 x

y

0 x

(a)

1 1

(b) y

0 x 1 _1

(c) y

0 x (2, 3)

6. (a) 7. (a)

(b) (b)

(c) (d) y

0 x 4

2

(e) y

0 1 x

(f ) y

0 1 x

(g) y

x 0

1 _1

(h) y

0 x 1

1

共 f ⴰ t兲共x兲 苷 4x² 8x  2

3, 3

共t ⴰ f 兲共x兲 苷 2x² 4x  5

y

0 x _1

1 共t ⴰ t ⴰ t兲共x兲 苷 8x  21

If you have had difficulty with these problems, you should look at Sections 1.1–1.2 of this book.

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xxii DIAGNOSTIC TESTS

D ANSWERS TO DIAGNOSTIC TEST D: TRIGONOMETRY

1. (a) (b)

2. (a) (b)

3.

4. (a) (b) (c)

5. (a) (b)

5兾3 兾10

150 2 cm

s3 ¹2 2

24 sin  24 cos 

共360兾兲 ⬇ 114.6

6.

8.

9.

1

15

(

)

0,

_π 0 π x

2 y

If you have had difficulty with these problems, you should look at Appendix A of this book.

D DIAGNOSTIC TEST: TRIGONOMETRY

1. Convert from degrees to radians.

(a) (b)

2. Convert from radians to degrees.

(a) (b)

3. Find the length of an arc of a circle with radius 12 cm if the arc subtends a central angle of .

4. Find the exact values.

(a) (b) (c)

5. Express the lengths and in the figure in terms of .

6. If and , where and lie between and , evaluate .

7. Prove the identities.

(a) (b)

8. Find all values of such that and .

9. Sketch the graph of the function without using a calculator.

300 18

5兾6 2

30

tan共兾3兲 sin共7兾6兲 sec共5兾3兲

a b 

sin x苷¹3 sec y苷⁵4 x y 0 2 sin共x  y兲

tan  sin   cos  苷 sec  2 tan x

1 tan²x 苷 sin 2x

x sin 2x苷 sin x 0 x  2

y苷 1  sin 2x a

¨ b 24

FIGURE FOR PROBLEM 5

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