冉 冊 冉 冊 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 fol- lowing 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.

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 simplfy.

(a) (b)

(c) (d)

(e)

4. Factor each expression.

(a) (b)

(c) (d)

(e) (f)

5. Simplify the rational expression.

(a) (b)

(c) (d)

y x  x

y 1 y  1

x x²

x² 4  x 1 x 2

2x² x  1

x² 9 ⴢ x 3 2x 1 x² 3x  2

x² x  2

x³y 4xy 3x^3兾2 9x^1兾2 6x^1兾2

x⁴ 27x x³ 3x² 4x  12

2x² 5x  12 4x² 25

共x  2兲³

共2x  3兲²

(

)(

)

共x  3兲共4x  5兲 3共x  6兲  4共2x  5兲

冉

冊

共3a³b³兲共4ab²兲² s200  s32

16^3兾4

冉

冊

5²³ 5²¹

3^4

3⁴ 共3兲⁴

D I A G N O S T I C T E S T : A L G E B R A

A

xxiv

6. Rationalize the expression and simplify.

(a) (b)

7. Rewrite by completing the square.

(a) (b)

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)

a兾x  b兾x 苷 1 a b 1

x y 苷 1 x  1

y

1 TC

C 苷 1  T

sa² b² 苷 a  b

sab 苷 sasb 共p  q兲²苷 p² q²

2x 3 x 1  1

ⱍ

ⱍ

x共x  1兲共x  2兲  0

x² 2x  8

4  5  3x  17

2x共4  x兲^1兾2 3s4  x 苷 0

3

ⱍ

ⱍ

x⁴ 3x² 2 苷 0

2x² 4x  1 苷 0 x² x  12 苷 0

2x

x 1 苷 2x 1 x x 5 苷 14 ¹2x

2x² 12x  11 x² x  1

s4 h  2 h s10

s5  2

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

共1, 4兴 共1, 7兲

共2, 0兲 傼 共1, 兲 共2, 4兲

关4, 3兲

12 5

2 3, ²²3

1, s2

1 ¹²s2

3, 4 1

6

2共x  3兲² 7

(

)

1 s4 h  2 5s2  2s10

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) 1 (d) 共x  y兲

x 2

x 1 x 3 x 2

x 2

xy共x  2兲共x  2兲 3x^1兾2共x  1兲共x  2兲

x共x  3兲共x² 3x  9兲 共x  3兲共x  2兲共x  2兲 共2x  3兲共x  4兲 共2x  5兲共2x  5兲

x³ 6x² 12x  8

4x² 12x  9 a b

4x² 7x  15 11x 2

x 9y⁷ 48a⁵b⁷

6s2

1 8 9

25 4

1

81 81

81

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

A N S W E R S TO D I AG N O S T I C T E S T A : A L G E B R A

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) x² y² 4 (f) 9x² 16y²苷 144 y x² 1 y 1 ¹2x

ⱍ

ⱍ

ⱍ

ⱍ

1  y  3

xy

AB AB AB

AB

B A B A B共5, 12兲

A共7, 4兲

x² y² 6x  10y  9 苷 0 共3, 2兲 共1, 4兲

2x 4y 苷 3 y

x

3

共2, 5兲 D I A G N O S T I C T E S T : A N A LY T I C G E O M E T RY

B

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

1. (a) (b)

(c) (d)

2.

3. Center , radius 5 4. (a)

(b) ; -intercept , -intercept

(c) (d) (e)

(f)共x  1兲² 共y  4兲²苷 100 3x 4y 苷 13

20

共1, 4兲 x 4 y ¹⁶3

4x 3y  16 苷 0

3⁴

共3, 5兲

共x  1兲² 共y  4兲²苷 52

y苷¹2x 6 x苷 2

y苷 5 y苷 3x  1

A N S W E R S TO D I AG N O S T I C T E S T B : A N A LY T I C G E O M E T RY

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

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) fⴰ t (b) tⴰ f (c) tⴰ t ⴰ t

t共x兲 苷 2x  3 f共x兲 苷 x² 2x  1

f f共1兲

f共2兲

f共x兲 苷

再

y苷 2^x 1

y苷 2sx y苷 sx

y苷 4  x²

y苷 共x  2兲³ 3 y苷 共x  1兲³

y苷 x³

y苷 f 共x  3兲  2 y苷 2 f 共x兲  1

y苷 f 共x兲

f

h共x兲 苷 s4  x  sx² 1 t共x兲 苷 s³x

x² 1 f共x兲 苷 2x 1

x² x  2

f共2  h兲  f 共2兲 f共x兲 苷 x³ h

f

f共x兲 苷 0 x

f共x兲 苷 2 x

f共2兲 f共1兲

f D I A G N O S T I C T E S T : F U N C T I O N S

C

6.(a) 7. (a)

(b) (b)

(c) 共t ⴰ t ⴰ t兲共x兲 苷 8x  21共t ⴰ f 兲共x兲 苷 2x² 4x  5

y

x _1 0

1

共 f ⴰ t兲共x兲 苷 4x² 8x  2

3, 3

(h) y

x 0 1

1

(g) y

x 0 _1 1

(f ) y

x

0 1

(e) y

x

0 1

(d) y

x 0 4

2

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. (c) y

x 0

(2, 3) y

x 0

(a) (b) y

1

1 0 x

1 _1

x 共, 1兴 傼 关1, 4兴 共, 兲

共, 2兲 傼 共2, 1兲 傼 共1, 兲 12 6h  h²

关3, 3兴, 关2, 3兴

2.5, 0.3

3, 1

2 y

0 x

1 1

FIGURE FOR PROBLEM 1

A N S W E R S TO D I AG N O S T I C T E S T C : F U N C T I O N S

If you have had difficulty with these problems, you should look at Sections 1.1–1.3 of this book.

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 y苷 1  sin 2xwithout using a calculator.

0 x  2 sin 2x苷 sin x

x 2 tan x

1 tan²x 苷 sin 2x tan  sin   cos  苷 sec 

sin共x  y兲  2

0 y

x sec y苷⁵4

sin x苷¹3

 b

a

sec共5兾3兲 sin共7兾6兲

tan共兾3兲

30 2

5兾6

18 300

6.

8.

9.

_π 0 π x

2 y

0,

1

15

(

)

1. (a) (b)

2. (a) (b)

3.

4. (a) (b) (c)

5. (a)24 sin  (b) 24 cos 

¹2 2 s3

2 cm

360兾 ⬇ 114.6 150

兾10 5兾3

D I A G N O S T I C T E S T : T R I G O N O M E T RY

D

a

¨ b 24

F I G U R E F O R P R O B L E M 5

A N S W E R S TO D I AG N O S T I C T E S T D : T R I G O N O M E T RY

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