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Practice Midterm I - 1

# Practice Midterm I - 1 - without the deFnition 4 Derive the...

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Name: Student number: 1. (10) Each part of the following question is worth 2 points. NO PARTIAL CREDIT will be given. (i) f ( x ) = x 1 - ln( x ) . Find f p ( x ). (ii) f ( x ) = x 2 sin(tan( x )). Find f p ( x ). (iii) State L’Hospital’s Rule. (iv) Find lim x 1 1 (1 - x ) 2 . (v) Sketch the graph of a function which satis±es lim x →±∞ f ( x ) = 0, f is increasing on the interval ( -∞ , 0), decreasing on the interval (0 , ), and concave down on the interval ( - 1 , 1). 1 continued on next page

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Name: Student number: 2. () Find all critical numbers for the function f ( x ) = x 2 e - x 2 . Use the ±rst or second derivative test to classify the critical numbers as local maxima or minima. 2 continued on next page
Name: Student number: 3. () Consider the function f ( x ) = b x 2 ln( | x | ) , if x n = 0; 0 , if x = 0. Use the deFnition of the derivative to Fnd f p (0). No credit will be given for attempting to Fnd the derivative

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Unformatted text preview: without the deFnition. 4. () Derive the quotient rule for derivatives from the product rule and the chain rule. 3 continued on next page Name: Student number: 5. () Consider the function f ( x ) = 1 + 1 /x + 1 /x 2 . (i) f p ( x ) =-1 x 2 (1+ 1 2 x ). Find the intervals on which f is increasing and the intervals on which f is decreasing. (ii) f pp ( x ) = 1 2 x 3 (1 + 1 3 x ). Find the intervals on which f is concave up and the intervals on which f is concave down. 4 continued on next page Name: Student number: (iii) Find lim x →∞ f ( x ) and lim x →-∞ f ( x ). (iv) Sketch the graph of y = f ( x ), incorporating all of the above information. 5 continued on next page...
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Practice Midterm I - 1 - without the deFnition 4 Derive the...

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