Looking for help on calculus

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Cevlna2088

Mathematics

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* 1 51% 10:28 PM < Scan_20180427_163455.jpeg 7. Use the connection between infinite series and improper integrals to determine which of the two infinite series converges, and which diverges (explain your reasoning): 1 1 Σ? vn 8. Sketch the curve defined by the polar equation r2 = cos 40 and find the area enclosed by the curve. 9. (a) Find an equation of the tangent plane to the surface z(x, y) defined by the equation 2 = 4,/xy at x = 1, y = 4. (Use partial derivatives zx and 2y.) (b) Use the equation of this tangent plane as a linearization of z to estimate z(1.1, 4.1). 10. Examine the three-dimensional surface described by the equation z = x + y2 – 3xy + 4 for local extrema and saddle points: find all its critical points and determine the type of each critical point. 11. Use multiple integration to find the average value of the function g(x, y) = y’e" on the triangle with vertices (0,0), (0, 4) and (4,4). Briefly explain the concept on which the definition of the average value of a function is based. Syzery Do simpler one first (inside) SS y el dx, dy priyanka patel 3h Final *Dil 51% 10:27 PM < Set 1.jpeg 1. An object moves along a coordinate line and its coordinate s depends on time t as s(t) = 213 - 1512 + 36t - 10. (a) Find the object's velocity v(t) at any given time t and sketch its graph. (b) Find the object's acceleration aft) at any given time t and sketch its graph. (c) When, if ever, does the body change direction? (Explain your opinion.) (d) Sketch a graph of the object's speed. When is it speeding up? Slowing down? Briefly summarize the principles you used to explore this motion process in a form that would be helpful to your students. 2x3 - x2 2. At which points is the rational function g(x) = continuous? x² – 2x-3 What kinds of discontinuities does it have? Give equations of all its asymptotes (vertical, horizontal, or oblique), find all its axis intercepts, and sketch its graph. Briefly summarize the principles you used to explore this function. 3. An object moving along a straight line is observed over the period of time from t = 0 to 1 = 20, and during this period its velocity as a function of t can be described by the formula v(t) = sin(t2/20). At the time 1 = 10, the coordinate of the object's location is 45. (a) Write a formula (in integral form) that describes location of the object s(t) at any time during the observation period te [0, 20). (b) Use your formula and the numerical integration procedure of your calculator to find the coordinate of the object at t= 8. (c) Briefly explain on what principles you based your solution. 4. (a) Find the volume of the solid obtained by rotating about the y-axis the region bounded by the graphs of y=e*, y=0, x = 0, and x = 1. (b) Examine the volume of the unbounded solid obtained by rotating about the y-axis the region between the graphs of y=e*, y=0, and x = 0. Is the volume finite or infinite? If it is finite, find it; if it is infinite, demonstrate it. 5. (a) Write a Taylor series expansion for the function f(x) = In(x) at x = 1. (b) Estimate the error of approximating In(x) with its fourth-order Taylor polynomial (with center x = 1) for xe (1.3, 1.6). (c) Determine for what values of x the series converges. 6. Use the precise definition of a limit to prove convergence of the sequence an = 4 – e?" (State what you intend to demonstrate to prove the existence of a limit, and then demonstrate it.) priyanka patel 3h Final
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Explanation & Answer

only Q 6 is left...i try and get back to you ...patel..please give some more time if you have

Question (1)
(a)
Object velocity  (t ) =

ds (t )
= 6t 2 − 30t + 36 = 6( x − 2)( x − 3)
dt

(b)
Object acceleration a (t ) =

d (t )
= 12t − 30 = 6(2t − 5)
dt

(c)
From (a) and (b), we can see that velocities are zero at t = 2,3 and acceleration is zero
at t = 2.5
Hence at t = 2 , a  0 ; thus, there is a change in direction.
Hence at t = 3 , a  0 ; thus, there is a change in direction.
(d)
Sl.No.
1.
2.
3.
4.

Time Duration
0-2 units
2-2.5 units
2.5-3 units
>3

Speed
Slowing down
Speeding up
Slowing down
Speeding up

Question 2:

g ( x) =

2x 3 − x 2
x 2 (2 x − 1)
=
x 2 − 2 x − 3 ( x − 3)( x + 1)

The function is discontinuous at x = −1,3 . The function doesn't approach a particular
finite value at x = −1,3 ; thus, the limit does not exist. This is an infinite discontinuity.

Question 3:
(a) Given that

=

 t2 
ds
= sin  
dt
 20 

 t2 
ds = sin  dt
 20 

Location of the object at point is given by:

 t2 
s(t ) =  sin  dt
 20 

(b)

8
 t2 
s(t = 8) =  sin  dt = 3.999
 20 
0
(c)

My solution is ba...


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