Tangent lines and areas between curves.

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The discussion centers on finding the tangent line to the function y=e^x at the point (1,e), with a proposed tangent line equation of x*e. To determine the area between this tangent line, the y-axis, and the curve y=e^x, the user equates the two functions and arrives at the equation (e^x)/(x)=e. Clarification is sought on solving for x, and the conversation shifts towards the necessity of integration for finding the area. The importance of understanding integration in this context is emphasized.
jason_r
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I have a function:
y=e^x
the tangent line at the point (1,e) would be x*e?

in order to find the area between the tangent line, the y-axis and y=e^x i equate these functions and solve for x?

I got this far

(e^x)/(x)=e
how do i solve for x
 
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jason_r said:
I have a function:
y=e^x
the tangent line at the point (1,e) would be x*e?

Correct!

in order to find the area between the tangent line, the y-axis and y=e^x i equate these functions and solve for x?

I got this far

(e^x)/(x)=e
how do i solve for x

Thats not exactly it. Do you know how to integrate?
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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