Download Are you ready for Calculus? Name: ________________________

Are you ready for Calculus?
Name: ________________________
Do ALL work on a separate sheet of paper.
1.
Simplify:
x  9x
a) 2
x  7 x  12
2.
2
3 2
b)
4
1 5
 2a 2 
2
a 
b
c)
3
 
a
3
9ab
b)
b
3
x1
 25
b)
ab  a
d) 2
b b
e)
a 1
b 1 a
2
 a 2 / 3   b3/ 2 
f)  1/ 2   1/ 2 
b  a 
1
 32 x 2
3
c)
log2 x  3
d) log 3 x  2 log 3 4  4 log 3 5
2
Simplify. Do not use a calculator!

2

b)
2log4 9  log2 3
2log 3 5
c) 3
Simplify:
1/ 2
a) log10 10
7.
1
1 3  5
Solve for x. Do not use a calculator!
a) log 2 5  log 2 x  1  log 2  x  1
6.
c)
p q
a) 5
5.
9  x 2
d)
3  x 1
Write each of the following expressions in the form ca b , where c, p and q are real numbers:
a)
4.
2
Rationalize the denominator:
a)
3.
1 1

x
5
c)
1 1

x 2 25
x  2x  8
b) 3
x  x2  2x
3
 1 
x 
 10 
b) log10 
c) 2 log
x  3log x1/ 3
Solve for the indicated variables:
a)
x y z
   1, for a
a b c
b)
c)
A  2 r 2  2 rh, for positive r
d) A  P  nrP, for P
e) 2 x  2 yd  y  xd ,
for d
f)
V  2  ab  bc  ca  , for a
2x 1 x

 0,
4
2
for x
Are you ready for Calculus?
8.
Page 2
For the following equations, complete the square and reduce to one of the standard forms
y  k  a  x  h  or x  h  a  y  k  :
2
a)
9.
y  x2  4x  3
6
9 y2  6 y  9  x  0
3
d) x  1
c) 8 x  27
2
4
3
4
b) 4 x  8 x  25 x  50  0
3
c) 8 x  27  0
2
3
Solve.
a)
3sin 2 x  cos2 x; 0  x  2
c)
tan x  sec x  2cos x;    x  
b)
cos2 x  sin 2 x  sin x;    x  
Without using a calculator, evaluate the following:
a)
b) sin
cos 210
e) cos
13.
c)
Solve over the real numbers.
6
12.
3x 2  3x  2 y  0
b) 4 x  8 x  25 x  50
4
a) x  16 x  0
11.
b)
Factor completely.
a) x  16 x
10.
2
9
4
f) sin
5
4
1
c)
3
2
tan 1 (1)
d)
sin 1 (1)
7
6
h)
cos 1 (1)
g) tan
Using the parent function y = sin(x), sketch each of the following graphs.
Do not use
your calculator.


a) y  sin  x 


4
c) y  2 sin x
 

2
b) y  sin 

d) y  cos x
2
e) y 
1
sin x
14.
Solve the equations:
a) 4 x  12 x  3  0
2
15.
b) 2 x  1 
5
x2
c)
x 1
x

0
x
x 1
Find the remainders after each division is performed:
a)
x5  4 x 4  x3  7 x  1 divided by x  2
b)
x5  x4  x3  2 x 2  x  4 divided by x3  1
Are you ready for Calculus?
16.
Page 3
a) The equation 12 x  23x  3x  2  0 has a solution x = 2. Find all other solutions.
3
2
b) Find all solutions for 12 x  8 x  x  1  0 knowing they are rational and between 1 and -1.
3
17.
Solve the inequalities:
a)
18.
x2  2 x  3  0
b)
2x 1
1
3x  2
c) x  x  1  0
2
Solve for x.
a)
19.
2
6
8

3
x x5
b)
5x  2  8
c)
2x 1  x  3
Determine the equations of the following lines:
a) the line through (-1, 3) and (2, -4)
b) the line through (-1, 2) and perpendicular to 2x – 3y + 5 =0
c) the line through (2, 3) and the midpoint of the segment from (-1, 4) and (3, 2)
20.
a) Algebraically find the point of intersection of the lines 3x – y – 7 = 0 and x + 5y + 3 = 0.
b) Graph the system of inequalities (remember to shade!) 3x  y  7  0
21.
and
x  5y  3  0 .
Find the equations of the following circles:
a) The circle with center (1, 2) that passes through the point (-2, 1).
b) The circle that passes through the origin and has intercepts equal to 1 and 2 on the x-axis and
y-axis, respectively.
22.
For the circle x
2
 y 2  6 x  4 y  3  0 , find:
a) the center and radius
23.
A circle is tangent to the y-axis at (0, 3) and has one x-intercept at (1, 0).
a) Determine the other x-intercept.
24.
b) the equation of the tangent line at (-2, 5)
b) Find the equation of the circle.
A curve is traced by a point P(x, y) which moves such that its distance from the point A (-1, 1) is three
times its distance from the point B (2, -1). Determine the equation of the curve.
Are you ready for Calculus?
25.
Page 4
a) Find the domain of the function f ( x) 
3x  1
x2  x  2
b) Find the domain and range of the following functions: i) f(x) = 7
26.
Graph
g ( x) 
ii) g ( x) 
5x  3
2x 1
x
.
x
a) Write this function as a piecewise function.
b) Find its domain and range.
27.
Simplify
f ( x  h)  f ( x )
where,
h
a) f(x) = 2x + 3
28.
b) f ( x) 
1
x 1
c)
f ( x)  x 2
The graph of y = f(x) is given as follows:
Graph the following functions.
a) f ( x  1)
c)
29.
b)
f ( x)  x( x 1)
a) The graph of a quadratic function has x-intercepts -1 and 3 and a range consisting of all numbers
less than or equal to 4. Determine the equation for this function.
y  2x2  4x  3 .
The following are parametric equations. Write as a single equation in x and y.
a)
32.
f ( x)
g ( x)  3x  2
b) Graph
31.
d)
Graph.
a)
30.
f ( x)
b) f ( x)
x  t 1

2
y  t t
 x  3 t  1
b) 
2
 y  t  t
c)
 x  sin t

 y  cos t
c)
f ( x)  x 2  2 x  1, x  0
Find the inverse of the functions:
a) f ( x)  2 x  3
b) f ( x) 
x2
5x 1
Are you ready for Calculus?
33.
A function f(x) has the graph to the right:
Sketch the graph of
34.
Page 5
f(x)
y=x
f 1 ( x) .
Express x in terms of the other variables in the picture.
a)
b)
x
r
h
x
h
r
t
t
35.
a) Find the ratio of the area inside the square but outside
the circle to the area of the square.
r
b) Find the formula for the perimeter of a window in the shape
of the picture below.
r
r
c) A water tank has the shape of a cone. The tank is 10 m high and has a radius of 3 m at the top. If
the water is 5 m deep (in the middle) what is the surface area of the top of the water?
d) A kite is 100 m above ground. If there is 200 m of string out, what is the angle between the string
and the horizontal? Assume the string is perfectly straight.
Are you ready for Calculus?
36.
Page 6
You should know the following trigonometric identities:
sin( x)   sin x
cos( x)  cos x
sin( x  y )  sin x cos y  cos x sin y
cos( x  y )  cos x cos y  sin x sin y
Use these to derive the following important identities, which you should also know.
a) sin x  cos x  1
b) sin 2x  2sin x cos x
2
2
c) cos 2 x  cos x  sin x
2
2
Answers. Remember, you will be held accountable for these concepts on Test #1. Come and get help if
necessary.
1. a)
x( x  3)
x4
b)
x4
x( x  1)
c)
2. a) 2 3  2 2 b) 1  5
c)
5x
5 x
7  2 15  3 3  5
11
2 2 1
ab
3
4. a) 1 b) -3/2 c) 8 d) 4/25, -4/25
5. a) log 2 5  x  1 b) log2 3 c) 25
3. a) 8a 6b 1
6. a)1/2
b) 3a1/ 2b3/ 2
b) -x
c)
d) ab 1 e) a 3/ 2b
f) a 5/ 6b1/ 2
c) log x 2
V  2bc
bcx
b) a 
2b  2c
cy  bz  bc

2x  y
e) d 
f) x 
1 
x  2y
7. a) a 
8. a) y  1   x  2 
2
9. a) x4  x  4 x  4
d)
3x  1
x
d)
c) r 
3
3
1
b) y     x  
8
2
2
2
2 h  4 2 h2  8 A
4
d) P 
2
1

c) x  10  9  y  
3

b)  2 x  5 2 x  5 x  2 c)  2 x  3  4 x 2  6 x  9 
 x  1 x  1  x 2  1
10. a) 0, 4, -4 b) 2, 5/2, -5/2 c) -3/2
 5 7 11
 5 

5
3
, ,
,
, ,
 2 n,
 2 ,
 2 n
11. a)
b)
c)
6 6 6 6
6 6
2
6
6
2
3
2
2
3



12. a) 
b) 
c) 
d) 
e)
f)
g)
h) 
4
2
2
2
2
3
3
13. graphs – verify your graphs on your calculator
3  6
14. a)
b) ½, -3
c) -1/2
2
A
1  nr
Are you ready for Calculus?
Page 7
a)-89 b) x 2
a) -1/4, 1/3 b) 1/3, -1/2 (double root)
a)  3,1 b)  , 2 / 3 1, ) c) all real numbers
a) 6, -10/3 b) 2, -6/5 c) 2, -4/3
7
3
19. a) y  3    x  1 b) y  2    x  1 c) y = 3
3
2
20. a) (2, -1) b) verify your graph with my key if you are unsure
15.
16.
17.
18.
21. a)
22. a)
 x  1   y  2   10
2
10
23. a) (9, 0)
2
2
1
5
2

b)  x     y  1 
2
4

1
 x  2
3
2
2
b)  x  5    y  3  25
b) y  5  
24. 8x 2  38x  9 y 2  20 y  43  0
25. a)
 , 2  1, 

b) i) D: all reals R:
7
ii) D:
 , 1/ 2 1/ 2, 
1, x  0
26. a) f ( x)  
b) D:  ,0 0,  R: 1,1
1, x  0
1
27. a) 2 b)
c) 2x + h
( x  h)( x  h  1)
28. verify your graphs with my key if you are unsure
29. graphs
30. a) y   x 2  2 x  3 b) graph
31. a) y  x 2  3x  2
b) y   x  1   x  1
6
3
c) x 2  y 2  1
x 3
x2
b) f 1 ( x) 
c) f 1 ( x)  1  x  2
2
5x  1
33. graph should be a reflection over the line y = x
rt  ht
rt
34. a) x 
b) x 
h
r 2  h2
32. a) f 1 ( x) 
4r 2   r 2
b) 4r  2 r
4r 2
36. derivations of identities
35. a)
c)
9
 m2
4
d) 30
R: all reals