Warm-up for Differential Calculus

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Nichols School Mathematics Department

Nichols School Mathematics Department

Summer Assignment

Warm-up for Differential Calculus

Who should complete this packet? Students who have completed Functions or Honors Functions and will be taking Differential Calculus in the fall of 2015.

Due Date: The first day of school

How many of the problems should I do? – ALL OF THEM

How should I organize my work? You should show all work in a separate sheets of loose-leaf paper. If a problem requires a graph, then you should use graph paper. Keep your materials in a folder with 3-prongs.

How will my teacher know that I’ve done the work? –Your teacher will collect your notebook on the first day of school. Your teacher may choose to QUIZ or TEST you on this material if he or she feels it is necessary – BE PREPARED!

How well should I know this material when I return? – You should recognize that you’ve seen this material before, and you should also be able to answer questions like the ones in this packet. If the material is revisited in your next class, it will only be for a brief amount of time – your teacher will assume that all you need is a quick refresher.

Note from your teachers:

We feel that this summer work will truly help you succeed this year. We understand that summer is a time for relaxation and fun, but it is imperative that you spend some time before you return reviewing your materials. This packet is mandatory, and you must treat it as you would any other extremely important homework assignment. You will be held accountable for this material. We also highly suggest that you do a bit of it at a time in the weeks leading up to school – don’t leave it for the last day!!!

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Warm-up for Differential Calculus

Instructions:

▪ Complete the problems on loose-leaf paper in pencil. If a problem requires a graph, please use graph paper and a ruler.

▪ At the top of each page of your work, write your name and the page number

▪ Complete all the problems carefully. Show enough work to indicate your method of solution.

▪ Make sure your work justifies your answer. ▪ Keep the packet and your work in a folder.

▪ Arrange your work in page order, and place the worksheets in report cover or 3-prong folder.

Remember, your teacher will collect your work on the first day of school! Late work will be severely penalized and it may NOT be accepted.

What if I get stuck? - You should check out additional study materials. Consult a standard precalculus textbook. Find a study buddy or classmate to help you remember the material. Consult the following websites for hints and examples:

http://coolmath.com/

http://www.math.ucdavis.edu/~marx/precalculus.html http://www.brightstorm.com/math/precalculus

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Warm-up for Differential Calculus page 1

1. Graph 2

( ) 2( 1) 4

f x   x  using transformations (shifting, stretching, reflecting, etc.) from the graph 2

yx . State the transformations needed. 2. For _{( ) 3} 2 ₁₂ ₄

f x  x  x :

i. Determine if the function has a maximum or a minima and explain how you know. ii. Algebraically, determine the vertex of the graph.

iii. Determine the axis of symmetry of the graph. iv. Algebraically, determine the intercepts of the graph.

v. Using the information in parts a-d, graph the function by hand. Plot additional points as needed in order to create an accurate graph.

3. Solve ₃ 2 ₂ _{1 0}

x  x  over the complex numbers.

4. Given the polynomial function _{( )} ₂₍ _{1) (}2 ₂₎

P x   x x , do the following: a. Find the x and y intercepts of the graph.

b. Determine whether the graph crosses or touches the x-axis at each x-intercept.

c. Describe the end behavior of the graph, that is, describe what happens to the y-values as x increases.

d. Using the information in parts a-c, make an accurate sketch of the graph.

5. For the polynomial function _{( ) 2} 3 ₅ 2 ₂₈ ₁₅

g x  x  x  x :

a. Determine the maximum number of real zeros that the function may have.

b. Given that 3 is a zero, determine the real zeros of g. Factor g(x) over the real numbers.

6. Give the equation of a polynomial function with real coefficients of degree 3 that has zeros of 5, 3 4i .

7. Sketch the graphs of these functions. Determine the amplitude, period, and any horizontal or vertical shifts. USE RADIANS.

a. 4 3sin 1 2 y  _ _   b. y 3 2cos 2 4        _ _  __    

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8. Give the equation of a sinusoid (sine or cosine function) that meets the conditions: a. has a maximum at (20,1) and a minimum at (25,-5)

b. has an amplitude of 3, a period of 180, and passes through the point (25,2)

9. Given, 3 4 and √ 1 , find the following and state its domain. a. (f g x)( ) b. f ( )x

g    

  c.



f g



( )x

10. Given the graphs of and , find (f g)( 1) , ( )(5)fg , and (g f)(4). 11. Let





2 6 ( ) 1 h x x 

 Given h x( ) ( f g x)( ) , name and .

12. Find the unknown value without using a calculator.

a. 3x343_b._{log 16 2}

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13. Solve each equation. Give an exact answer when possible. When using a calculator, round your answer to three decimal places.

a. ₅x2 ₁₂₅_b._log



₉



₂ x  c. 8 2 x 4 e   d. _log



2 ₃



_log



₆



x   x e. ₇x3 ₂x_f.









2 2 log x4 log x4 3

14. Given log 2 0.301_a  and log 3 0.477_a  , find each of the following:

a. log 6_a b. log_a 3 c. log 2

3

a

   

  d. log 36a

15. Use transformations to graph each function: identify the “base” function, then describe any shifts, stretches, etc. Sketch an accurate graph of the function. Determine the domain and give the equation of any asymptotes.

a. _{( ) 4}x 1 ₂

f x    b. g x( ) 1 log  ₅



x2



16. A 50 mg sample of a radioactive substance decays to 34 mg after 30 days. a. How much of the substance is left after 10 days?

b. How long will it take for there to be 2 mg remaining?

17. Find the inverse of ( ) 2 3

f x x

 .

18. If the point (3,-5) is on the graph of the function g x( ), what point must be on the graph of 1_{( )}

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19. Each of the following graphs passes through the point P(1, 2). Match the correct function to

each graph. Identify a characteristic that enables you to determine which graph.

a. 2x y b. y2sin(30 ) 1x  c. ylog₂x d. 1 3 2 2 y x e. 2 ₄ ₅ yx  x

20. Find the domain of ( ) ₂ 1

11 10 x f x x x     , 1 ( ) g x x  and



g f



( )x . 21. Simplify: 2 ₈ ₁₆ 3 4 x x x    22. Multiply: 2 2 6 3 9 2 4 x x x x      23. Simplify: 1 1 x y x y  

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24. For each angle on the unit circle shown, fill in the angles (in degrees and radians). You may write your work on this sheet. Be sure to turn it in with your other work.

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Nichols School Mathematics Department 25. Give the exact value of each of the following.

a. sin 3 2        b. 2 cos 3  _      c. sec(45 ) d. cot(215 ) e. tan 7 6       

26. Find all values of  on [0, 2 ) for which the following are true.

a. cos 3

2

   b. sin 2

2

  c. cot  0

27. Verify each identity. Remember to work only on one side of the equation.

a. sin tan  cos sec b. tan cot _{1 2cos}2

tan cot

  _

 

 _{ }



28. Solve each equation on the interval 0  2

a. _4sin2__{ }_{3 0}

c. sin 3

 

   1 0

b. _2cos2___cos_ _₁