Calculus with Applications Summer Review Packet
This packet is a review of the entering objectives for Calculus/Applications and is due on the first
day back to school. Please show all work NEATLY and on a SEPARATE sheet of paper.
Have a great summer!!
1. Simplify. Show work that leads to your answer.
a)
x−4
2
x − 3x − 4
x3 − 8
x−2
b)
c)
5− x
x 2 − 25
2. Trigonometric Pythagorean Identities: a) _______ b) _______ c) _______
3. Simplify each expression. Write answers with positive exponents where applicable:
1
1
a)
−
x+h x
1
f) log
100
2
2
b) x
10
x5
g) ln e
c)
7
2
12 x −3 y 2
18 xy − 1
h) 27
2
3
5
3
3
d) (5a 3 )(4a 2 )
e) (4a 3 ) 2
I) log 1 8
j) ln 1
2
4. Solve 4x + 10yz – 3 = 0 for z
5. Given: f(x) = {(3, 5), (2, 4), (1, 7)}
a) h(g(x))
g ( x) = x − 3
h(x) = x 2 + 5 find, simplified:
c) f ’(x) (inverse)
b) g(h(-2))
d) to find g’(x) (the inverse) you switch _____ and then solve for _____. Fill in the blanks and then
find the inverse.
5
6. Expand and simplify:
∑ 3n − 6
n=2
(remember
∑
means sum)
7. Using EITHER the slope/intercept (y = mx + b) or the point slope ( y − y1 = m( x − x1 )) form of a line,
write an equation for the lines described: SHOW ALL WORK
a) with slope -2, containing the point (3, 4)
b) containing the points (1, -3) and (-5, 2)
c) with slope 0, containing the point (4, 2)
d) parallel to 2x – 3y = 7 and passes through (5,1)
e) perpendicular to the line in problem #7 a, containing the point (3, 4)
8. Without a calculator, determine the exact value of each expression:
a)
sin
e) cos
π
2
π
3
b) sin
3π
4
c) cos π
f) tan
7π
4
g) tan
2π
3
9. For each function: Make a neat sketch, putting numers on each axis
For #d, e, f and g also write the equations of the asymptotes
a) y = sin x
d) cos
h) tan
7π
6
π
2
Name the Domain and Range
b) y = x 3 − 2 x 2 − 3x
c) y = x 2 − 6 x + 1
f) y = e x
d) y =
x+4
x −1
e) y = ln x
g) y =
1
x
h) y =
j) y = x + 3 − 2
k) y =
x2 − 4
x+2
x−4
i) y = 3 x
l) y =
x2 + 4
10. Make a neat sketch of the piecewise function:
x 2 if x < 0
y = x + 2 if 0 < x < 3
4 if x > 3
11. Solve for x, where x is a real number. Show the work that leads to your solution:
a) 2 x 2 + 5 x = 3
b) ( x − 5) 2 = 9
c) (x + 3)(x – 3) > 0
d) log x + log(x – 3) = 1
e) x − 3 < 7
f) ln x = 2t – 3
g) 12 x 2 = 3 x
h) 27 2 x = 9 x −3
i) x 2 − 2 x − 15 < 0
12. Determine all points of intersection:
a) parabola y = x 2 + 3x − 4 and the line y = 5x + 11
b) y = cos x and y = sin x in the first quadrant