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math104-f21:hw13 [2021/12/03 20:42]
pzhou 		math104-f21:hw13 [2026/02/21 14:41] (current)
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 ====== Solution ====== ====== Solution ======
-====== HW 13 ====== 
-Due Monday (Nov 29) 9pm. 8-O problem 6 contains a typo, and it is updated now.  
  
 1. Assume that $\lim_{x \to 0} f'(x) = 1$. Prove that $f'(0)=1$. (Hint, you can use mean value theorem, and definition of the $f'(0)$. ) 1. Assume that $\lim_{x \to 0} f'(x) = 1$. Prove that $f'(0)=1$. (Hint, you can use mean value theorem, and definition of the $f'(0)$. )
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 3. Ross Ex 29.3 3. Ross Ex 29.3
  
-{{:math104:pasted:20211203-122902.png}}+{{math104-f21:pasted:20211203-122902.png}}
  
 Consider the interval $[0,2]$, the slope of the segment $\frac{f(2)-f(0)}{2-0} = 1/2$, hence there is an $x_1 \in (0,2)$, such that $f'(x_1)=1/2$. Consider the interval $[0,2]$, the slope of the segment $\frac{f(2)-f(0)}{2-0} = 1/2$, hence there is an $x_1 \in (0,2)$, such that $f'(x_1)=1/2$.
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 4. Ross Ex 29.5  4. Ross Ex 29.5 
  
-{{:math104:pasted:20211203-123421.png}}+{{math104-f21:pasted:20211203-123421.png}}
  
 We can prove that $f'(x)=0$ for all $x$, indeed, we have We can prove that $f'(x)=0$ for all $x$, indeed, we have
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-{{:math104:pasted:20211203-123645.png}}+{{math104-f21:pasted:20211203-123645.png}}
  
 (a) Taking derivatives once both upstairs and downstairs, we get  (a) Taking derivatives once both upstairs and downstairs, we get 
math104-f21/hw13.1638564164.txt.gz · Last modified: 2026/02/21 14:43 (external edit)

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