Summer Packet - Calculus HR - Mrs. Reilly
Welcome to Calculus Honors! This class contains a mixed bowl of concepts intertwined
with the new concepts to be learned. This packet serves as a review of material that
serves as the foundation needed in order to be successful in Calculus.
This packet contains a majority of what must be known in order to achieve the most
success possible next year. This entire packet is not mandatory however you will be
assessed on the information in here during the first week and a half of school. This
grade will be your first test grade of term 1.
You should be comfortable leaving your answer exact (no rounding needed...you can leave e
or pi in your answer, etc.) or rounded to any decimal asked. We often round to the
thousandth place in Calculus.
I periodically check my email over the summer. If you have any questions, feel free to
drop me an email at: ​
[email protected]​
or ​
[email protected]
I.
II.
III.
IV.
INDEPENDENT MUST KNOWS
Algebra
A. Exponents (operations with integer, fractional, and negative exponents) ​
*
B. Factoring (GCF, trinomials, difference of squares and cubes, sum of cubes, grouping) ​
*
C. Rationalizing (numerator and denominator)
D. Simplifying rational expressions ​
*
E. Solving algebraic equations and inequalities (linear, quadratic, higher order using synthetic
division, rational, radical, and absolute value equations, discriminant and what it means
about the solutions of a quadratic equation) ​
*
F. Simultaneous equations (systems) *​
Graphing and Functions
A. Lines (intercepts, slopes, write equations using point-slope and slope intercept form,
parallel, perpendicular, distance and midpoint formulas) *​
B. Functions (definition, notation, domain, range, inverse, composition, intercepts, end behavior)
*
C. Basic shapes and transformations of the following functions (absolute value, rational, root,
higher order curves, log, ln, exponential, trigonometric, piecewise, inverse functions) *​
D. Tests for symmetry: odd, even (both algebraic and graphical)
E. Interval notation*
Geometry
A. Pythagorean theorem *​
B. Area formulas (circle, polygons, surface area of solids)
C. Volume formulas
D. Similar triangles
Logarithmic and Exponential Functions
A. Simplify expressions (use laws of logarithms and exponents) *​
B. Solve exponential and logarithmic equations (includ ln as well as log) *​
C. Sketch graphs *​
D. Inverses ​
*
V.
Trigonometry
A. Unit circle (definition of functions, angles in radians and degrees) ​
*
B. Use of Pythagorean identities and the sum/difference identities)
C. Solve equations ​
*
D. Right triangle trigonometry ​
*
E. Graphs ​
*
* A​
solid​
foundation in these are extremely important!
STUDENT RESPONSIBILITY
1. Know where to find your homework. It may not be stated during class however will always be
online.
2. Completion of homework daily, even though it does not count as part of your grade
directly...without this success will not be obtained!
3. Check your answers for homework and ask in class or after school to review any that you
may not understand. We will not always get to review homework during class time.
4. Know your limit...when should you stay after and seek extra help?
5. Quick paced class, college level material, so keeping pace is very important.
6. Take notes every day.
7. Do not memorize math...you understand it
Practice Problems.
A. Simplify. Leave your answers with positive exponents only.
​
B. Factor Completely.
-3​
-5​2​
2​
3
x​
- x​
y​+ x​
y - y​
C. Rationalize what is being asked.
denominator
denominator
4
√5
5+x
√x+1
numerator
√3x+1−√x
x−2
D. Simplify the rational expression.
Solve the equation.
E. Factor each of the following first by finding the possible zeros (rational root theorem)
and then find the zeros of p(x), analytically.
​
​
p(x) = x3​
+ 4x2​
+ x - 6
​
​
p(x) = x3​
+ 5x2​
- 2x - 24
​
​
p(x) = x3​
- 6x2​
+ 3x + 10
​
​
p(x) = x3​
+ 2x2​
- 19x - 20
​
​
​
p(x) = x4​
+ 5x3​
+ 6x2​
- 4x - 8
​
​
​
p(x) = x4​
+ 11x3​
+ 41x2​
+ 61x + 30
​F. Solve the systems of equations.
G. Lines
Write an equation for the line through the
points (1, 1) with a slope of 2. Write this in
point-slope form, slope-intercept form and
standard form.
Let ​
L​
be 2x + y = 4. Write an equation for
the line through point P(-2, 2) that is parallel to
line ​
L​
and another equation for a line that is
perpendicular to line ​
L​
.
For what value of ​
k​
are the two lines
2x + ky = 3 and x + y = 1 parallel?
perpendicular?
Graph this piecewise function.
Graph this absolute value function.
H. Functions
a. How can you identify an equation as a function?
b. How can you identify a graph as a function?
c. Given the information below find the inverse/composition and the domain of each.
​
f(x) = x2​
+ 3x - 2
g(x) = 4x - 3
h(x) = ln(x)
g-1​​
(x)
h-1​​
(x)
​
w-1​
(x) for x >= 4
f(g(x))
h(g(f(1)))
w(x) = √x − 4
I.
State the domain of the function provided. If there is a discontinuity, state the type:
removable or infinite (asymptote).
J. Write the following absolute value equations into piecewise functions.
K. Solve the following inequalities.
​
x2​
+ 6x < 0
L. Function transformations.
​
If ​
f(x) = x2​
- 1​
, describe in words what the following would do to the graph of ​
f(x)​
.
f(x) - 4
f(x - 4)
-f(x + 2)
5f(x) + 3
f(2x)
|f(x)​
| Given the graph of f(x), sketch the graphs below.
M. Show your work, analytically, to state whether the relation is even, odd or neither.
​
f(x) = 2x2​
- 7
​
f(x) = -4x3​
- 2x
​
f(x) = 4x2​
- 4x + 4
f(x) = x − 1x
f (x) = |x| − x2 + 1
y = ex − e1x
N. Solve each equation. Remember...if you can tell if a quadratic factorable if the
discriminant is equal to a perfect square.
​
7x2​
- 3x = 0
4x(x - 2) - 5x(x - 1) = 2
​
x2​
+ 6x + 4 = 0
​
2x2​
- 3x + 3 = 0
​
2x2​
- (x + 2)(x - 3) = 12
x + 1x = 13
6
​
​
x4​
- 9x2​
+ 8 = 0
x − 10√x + 9 = 0
1
x2
− 1x = 6
​
8x4​
= x
O. Find the equations of the vertical, horizontal and slant asymptotes (if they exist).
P. Find the exact value of the unknown.
(y in terms of x)
Q. Word Problems. Please use your calculators to aid in your work.
​
a. A cannery will package tomato juice in a 2-liter (2000 cm3​
) cylindrical can. Find the
​
radius and height if the can is to have a surface area that is equal to 1000 cm2​
.
b. A gym wants to build a rectangular swimming pool with the top of the pool having
​
a surface area of 1000 ft2​
. The pool is required to have a walk of uniform width
2ft. surrounding it. Let ​
x​
be the length of one side of the swimming pool.
i. Express as a function of ​
x​
the area of the plot of land needed for the pool
and surrounding sidewalk.
ii. Find the dimensions of the plot of land that has the least area. What is
the least area?
c. A little league team has a throwing machine that propels a baseball upward from
ground level with an initial velocity of 55 ft/sec.
i. Does the ball reach a height of 52ft?
ii. How many seconds after the ball is propelled is it 42ft. above the ground?
d. From the top of the 100-foot tall building a man observes a car moving toward
the building. If the angle of depression of the car changes from 22 degrees to
46 degrees during the period of observation, how far does the car travel?
e. You are enclosing a garden in the shape of a right triangle, by a using 56 feet of
fence. If one leg is 24 feet, what are the dimension of the other two sides?
R. Average Rate of Change
1. State the average rate of change between the points (6, -2) and (-1, 12)
2. A cyclist travels 12 miles from home to work. She leaves at 7:02 am and arrives at 7:47
am. Find her average rate of change (speed) in miles per hour.
3. Find the average rate of change of the function y = √x over the interval [0, 16].
4. Find the average rate of change between the endpoints of the function graphed below.
S. Graphs. Sketch a possible graph of a function with the following information.
1. The function has a positive slope on the interval (-∞, 5) and a negative slope on the
interval (5, ∞).
2. The function has a negative slope on the interval (-∞, -3) U (3, ∞) and a constant slope
on the interval (-3, 3).
3. The function has a positive slope on the interval (-∞, 0) U (0, ∞), an undefined slope at
x = 0, and a horizontal asymptote at y = 0.
4. Match each story to the most appropriate graph. The x-axis represents time and the
y-axis represents distance.
a. I started on my way to school then realized I forgot my paper. I went back to
pick it up and then went back to school.
b. I started on my way to school then realized I was going to be late so I sped up.
c. I started on my way to school then got stopped for speeding. After I received
my ticket, I continued on my way to school.
NOw match the a, b and c from above with the new graphs below. The x-axis still
represents time and the y-axis now represents velocity.
T. Evaluate using the table and/or graph provided.
h(0)
g(3)
h(x) = 3
5g(-2) + h(-3)
h(g(-1) - 5)
h(x) = g(9) + 6
g(x) = -1
g(h(4))
Solutions will be posted on this link over the summer.
You can check the following websites for the solutions:
http://franklinhighschool.weebly.com/​
& ​
solutions
http://franklinhigh.vt-s.net/Pages/FranklinHS_Departments/Math/S0173CDD5-01F6C877