Calculus question help

[Destiny of Pi]

Hey guys, 

I was sick and missed some math classes. Here are two questions that I can partially solve them but not for the second part. It would be great if some of you could help me with the steps and process. 

For this question, I got that when x=0.4, y=1.573, but I don't know how to justify whether the approximation if greater or smaller than the actual value. 

For this question, I found that the Maclaurin series is ln(2) +1/2x+1/8x^2. I wonder how could we use this to solve part b. 

[kw0573]

1b. Because slope increases with x and y, and you are always underestimating the derivative, you would be underestimating the value at x = 0.4

2b Limits with indeterminate form (0/0, infinity/infinity etc) can be evaluated by substituting expansions at the desired x value for both numerator and denominator. For example, limit of sin x / x as x approaches 0 can be evaluated as
i) sin x expand at x = 0, is x - x^3 / 6 + ..., x is just x

ii) (x - x^3 /6 + ...) / x  = x / x - x^3/6 / x + ... = 1 - x^3/6 /x + ..., which when evaluated at x = 0, is just 1.

iii) hence for sin x / x at x -->0 the limit is 1. You can do a similar analysis for the desired expression.

[Destiny of Pi]

Thanks, I have another question. 

For this question part b, I can use a graphing calculator. But I don't know how to find the horizontal asymptote to the function. The function is f(x)= x^(1/x)

[kw0573]

Horizontal asymptotes only at x --> ± infinity. Because f(x) is not real in x <0, x --> + infinity gives asymptote at f(x) = 1. You should get f(x) = 1 as horizontal asymptote from graphing calculator. An alternative (more rigorous) way is to transform the function.

define g(x) = ln (f(x)) = 1/x * ln x

We know that the limit as g(x) of x approaches infinity is 0

f(x) = e^(g(x)), same limit would evaluate to e^0 = 1

Edited March 5, 2018 by kw0573

[Destiny of Pi]

Thanks! I just have one more question

For this question, I got the slopes are all zero for part a and b. For part c, isn't the slope just y=3? since it passes through (0,3)? How is it 8 marks?

[kw0573]

x and y with 5 values each create 25 (x,y) pairs.

So for example the slope at (0,3) is -3.

[Destiny of Pi]

3 minutes ago, kw0573 said:

x and y with 5 values each create 25 (x,y) pairs.

So for example the slope at (0,3) is -3.

sorry, my bad

Then how do you do part c?

[kw0573]

You would use the method of integrating factor (which should be in your textbook). You can also find it online. It's just an application of both e^x derivatives and product rule.

Edited March 6, 2018 by kw0573
not chain rule

[Destiny of Pi]

ok thanks!!!