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Unformatted text preview: Math 34A Practice Final Solutions Fall 2007 Problem 1 Find the derivatives of the following functions: 1. f ( x ) = 3 x 2 + 2 e 3 x 2. f ( x ) = x 2 +1 x 3. f ( x ) = ( x + 2 a ) 2 4. Is the function 3 t 2 4 t t 3 increasing or decreasing when t = 1? 5. Find a nonzero function f ( x ) such that f ( x ) 4 f ( x ) = 0. Solution 1. f ( x ) = 6 x + 6 e 3 x 2. f ( x ) = 1 1 x 2 3. f ( x ) = 2 x + 4 a 4. f ( x ) = 6 t 4 3 t 2 . So, f (1) = 6 4 3 = 1 < 0, which means f is decreasing. 5. f ( x ) = e 4 x Problem 2 (a) What is the equation for the line tangent to f ( x ) = √ x at x = 1? Solution First, we find the slope of the line by finding f (1): f ( x ) = 1 2 √ x ⇒ f (1) = 1 2 Next, we find a point on the line by substituting x = 1 into f ( x ): (1 ,f (1)) = (1 , 1) Then we can use pointslope format to find the equation of the line: y 1 = 1 2 ( x 1) ⇒ y = 1 2 x + 1 2 (b) Use the tangent line approximation at x = 1 of f ( x ) to approximate √ 1 . 1. Solution To do this, we simply substitute x = 1 . 1 into the equation for line tangent to f ( x ) at x = 1: y = 1 2 (1 . 1) + 1 2 ⇒ y = 1 . 05 1 Problem 3 If the half life of some element is 20 years, how long does it take until 1% of the element remains? Solution Call the initial starting amount of the element A . Then according to the half life formula, the amount left of the element at year t is given by A ( t ) = A 1 2 t/ 20 Then to find when 1% of the element is left, we set A ( t ) equal to 0 . 01 · A , and solve for t : . 01 · A = A 1 2 t/ 20 log 0 . 01 = t 20 log 0 . 5 t = 20 log 0 . 01 log(0 . 5) Problem 4 A jet airliner flies at 300 mph for the first half hour and last half hour...
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 Fall '07
 RickRugangYe
 Derivative

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