Math

h.w_26.docx

1. Use limit Laws to find: (a) imit as (n to infinity) [n^2-1]/[n^2+1] (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 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 = 1 , discontinuous from the right at = 1 , and such that (1) = 2 .

4. Determine all horizontal asymptotes of (x) = [x-2]/[x^2+1]+2

5. Determine all vertical asymptotes of (x) = [x-2]/[x^2-11]+2

6. Which of the functions do not have any vertical nor horizontal aysmptotes? (a) in x (b) (c) (d) n x (e) ^(-1)

7. Differentiate: (a) in (x^2) (b) in^2 x (c) ^(1/x) (d) n [x-1]/[x^3+1] (e) os (cubic root of (x))

8. Find y/dx : (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 ed y = f(x) and lue y = g(x) are given. Find (a) f g)'|_(x = 2)

(b) f/g)'|_(x = 2)

ttp://calculus.sfsu.edu/latexrender/pictures/0050abe9c5c53b93be4a53f787e4ccff.png

10. (i) Find critical numbers of the function. Then determine if the critical number is a local minimum, local maximum, or neither. (a) (x) = x^2-3 (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) (x) = x^2-3 ( 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 = e^(5 x) at 0,1) .

13. Find an equation of the tangent line to the graph of ^2+y^2 = 25 at 3,4) .

14. Find intervals of concavity. (a) (x) = x^2-3 ( 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 x-2 y = 9 that is closest to the point 0,1) (Check your answer with the grapher)

17. Find the general antiderivative of (a) (x) = 9 , (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) ntegral from 2^4 t^(-3) dt . (b) Find ntegral from (-2)^0(e^x+x) dx . (c) Find ntegral from 2^(p) t dt (the answer is a function of ).

19. State the Differentiation Part of the Fundamental Theorem of Calculus. Then find a /dx integral from 2^x cos (t^4) dt .

b Find /dx integral from x^6 cos (root (s^4+1)) ds . 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) = (x-2)(x+3)(x+5) (b) = [x-1]/[3 x+4]

21. The graph = f(x) 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 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 ? You can assume that '(x) is continuous. In your explanation use the The Mean Value Theorem, to argue that for some , '(c) = 1 . Then use the Intermediate Value Theorem applied to '(x) to argue that for some , '(d) = 0 .

22. [If you are not allowed to drink alcohol, please replace the word beer with tea] Let (t) , 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 (x) = 1/2(1-x)(x^2-x-2) is illustrated below.

(a) Find the and intercepts.

(b) Determine the intervals on which is decreasing and intervals on which it is increasing.

(d) Determine concavity intervals for . Determine inflection points on the graph .

(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 mileshour and the other half time with speed 30 mileshour. Cyclist B rode half the total distance with speed 10 mileshour and half the distance with speed 30hours. Sketch a graph of the distance function for both cyclists. Calculate the distance covered by each cyclist.