Math

1. Use limit Laws to find: (a)

    (b)

imit as (n to-infinity) [n-1]/[n^2+1]

    (c)

imit as (x to 2) x^4-2+sin (x pi)

2. What does it mean that a function

is continuous at

= 1

from the left? Explain why the function

(x) = |x-1|

is continuous at

= 1

from the left. Is

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continuous from the right?

3. What does it mean that a function

is differentiable at

= 1

? Is

(x) = |x-1|

is differentiable at

= 1

?

Sketch a graph of a function that is continuous from the left at

, discontinuous from the right at

= 1

, and such that

(1) = 2

.

4. Determine all horizontal asymptotes of

5. Determine all vertical asymptotes of

6. Which of the functions do not have any vertical nor horizontal aysmptotes? (a)

    (b)

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    (c)

^x

    (d)

n x

    (e)

^(-1)

   

7. Differentiate: (a)

    (b)

in^2 x

    (c)

^(1/x)

    (d)

n [x-1]/[x^3+1]

    (e)

os (cubic root of (x))

   

8. Find

: (a)

= x^2

    (b)

+x = y^2

    (c)

= x^x

    (d)

-x = sin (x y)

    (e)

= (x-1)(x+3)(y-1)

   

9. Graphs of

and

lue y = g(x)

are given. Find (a)

f g)'|_(x = 2)

(b)

(c)

10. (i) Find critical numbers of the function. Then determine if the critical number is a local minimum, local maximum, or neither. (a)

    (b)

(x) = x^3-x^2+x-3

    (c)

(x) = (x-2)(x+4)

    (d)

(x) = sin x

(

< x < 2 pi

)     For the functions above determine intervals on which they are increasing and intervals on which they are decreasing.

11. Find the absolute maximum and absolute minimum of the functions. (a)

(

less or = x less or = 2

)     (b)

(x) = x^3-x^2+x-3

(

3 less or = x less or = 3

)    (c)

(x) = (x-2)(x+4)

(

5 less or = x less or = 3

)     (d)

(x) = 2 sin x

(

< x < 2 pi

)    

12. Find an equation of the tangent line to the graph of

at

0,1)

.

13. Find an equation of the tangent line to the graph of

at

3,4)

.

14. Find intervals of concavity. (a)

(

less or = x less or = 2

)     (b)

(x) = x^3-x^2+x-3

(

3 less or = x less or = 3

)    (c)

(x) = (x-2)(x+4)

(

5 less or = x less or = 3

)     (d)

(x) = 2 sin x

(

< x < 2 pi

)    

15. Find two numbers whose sum is

and whose product is a maximum.

16. Find the point on the line

that is closest to the point

0,1)

(Check your answer with the grapher)

17. Find the general antiderivative of (a)

, (b)

(x) = x^(-1/3)+1

, (c)

(x) = e^(x)+cubic root of (x)

, (d)

(x) = sin x+3 cos x

.

18. State the Evaluation Part of the Fundamental Theorem of Calculus. Then find the definite integrals: (a)

.    (b) Find

ntegral from (-2)^0(e^x+x) dx

.    (c) Find

ntegral from 2^(p) t dt

(the answer is a function of

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).  

19. State the Differentiation Part of the Fundamental Theorem of Calculus. Then find a

.   

b Find

.    c Find

/dx integral from 2^(2 x+1) ln (t+1) dt

.    d Find

/dx integral from (-x)^x[z+1]/[z+2] dz

.    e Find

/dx integral from (-3x)^(2) 2^(t^2) dt

.  

20. Use graph sketching techniques to graph: (a)

(b)

= [x-1]/[3 x+4]

21. The graph

passes through the points

1,5)

and

3,7)

. The tangent line to

= f(x)

at

3,7)

has the equation:

=-2 x+13

[earlier there is a typo here]. Sketch a possible graph of

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and the tangent line. What is the average rate of change of

(x)

on the interval

leq x leq 3

? What is the instantaneous rate of change of

(x)

at the point

3,7)

? Explain. Explain why

(x)

has a critical number in the interval

leq x leq 3

? You can assume that

'(x)

is continuous. In your explanation use the The Mean Value Theorem, to argue that for some

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,

'(c) = 1

. Then use the Intermediate Value Theorem applied to

'(x)

to argue that for some

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,

'(d) = 0

.

22. [If you are not allowed to drink alcohol, please replace the word beer with tea] Let

,

less or = t < 24

, denote the number of pints drunk in the previous hour. For example, if between 21.00 and 22.00 you had two pints of beer, then

(t) = 2

when

in [22,23)

. Sketch a plausable graph of

(t)

. (a) Explain what

ntegral from (0)^(23) f(t) dt

means. (b) What does the function

(x) = integral from (0)^(x) f(t) dt

represent?

23. The graph of

is illustrated below.

(a) Find the

and

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intercepts.

(b) Determine the intervals on which

is decreasing and intervals on which it is increasing.

(c) Find local minima and local maxima of

(x)

.

(d) Determine concavity intervals for

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. Determine inflection points on the graph

= f(x)

.

(e) Determine (vertical and horizontal) asymptotes if any?

24. [optional] Cyclists A and B rode along a straight one way road starting from the same location and they both rode for an hour. Cyclist A rode half the time with speed 10 miles

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hour and the other half time with speed 30 miles

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hour. Cyclist B rode half the total distance with speed 10 miles

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hour and half the distance with speed 30

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hours. Sketch a graph of the distance function for both cyclists. Calculate the distance covered by each cyclist.