MHB Find the equation of the tangent line (Can someone check my work?)

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The discussion centers on finding the equation of the tangent line to the function g(x) = arctan(x) + ln(x) at x = 1. The provided equation, y - π/4 = 3/2(x - 1), is confirmed to be correct, but the lack of detailed work raises concerns about the validity of the solution. Participants emphasize the importance of showing the derivative g'(x) and the formula for the tangent line to ensure understanding and correctness. Without this supporting work, it is difficult to verify the accuracy of the answer. Overall, demonstrating the process is deemed more crucial than simply providing the final answer.
shamieh
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Find the equation of the tangent line to g(x) = arctan(x) + ln(x) @ x = 1

$$y - \frac{\pi}{4} = \frac{3}{2}(x - 1)$$
 
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Looks good to me. :D
 
shamieh said:
Find the equation of the tangent line to g(x) = arctan(x) + ln(x) @ x = 1

$$y - \frac{\pi}{4} = \frac{3}{2}(x - 1)$$

Your answer is indeed correct, but as no actual work is shown, we cannot verify that you didn't just happen to "guess" the right answer (although you probably didn't).

Things a teacher might want to see included in your answer:

1) What is g'(x)?

2) What is the formula for the tangent line to g(x) at x = a?

The correct two answers to the questions I have listed above are actually more important than "the final answer".
 
Deveno said:
Your answer is indeed correct, but as no actual work is shown...

This is an excellent point. If you answer happened to be incorrect, we would not have been able to address where you made any errors.
 

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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