P MT 12 sln

# P MT 12 sln - N be to guarantee that | f x-3 |< ε for...

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Math 1A Calculus Fall, 2004 Prof. Haiman First Midterm Exam—Solutions 1. Find the domain of the function f ( x ) = x - 3 - x . Both x and 3 - x must be positive, so the domain is [0 , 3]. 2. Find a formula for the inverse function g ( x ) of the function f ( x ) = e x 2 +1 . Sorry, this question was badly formulated. I should have speciﬁed that the domain of f ( x ) is [0 , ). Since I didn’t, you could correctly answer that the function is not one-to-one and therefore has no inverse. If you assume that the domain of f ( x ) is meant to be [0 , ), then solving y = e x 2 +1 for x gives x = p ln( y ) - 1, so g ( x ) = p ln( x ) - 1. (The domain of g ( x ) is [ e, ), but it was not required that you specify it.) 3. Evaluate the limit, if it exists (possibly as an inﬁnite limit). lim x 3 - x - 5 x - 3 . The limit is + , since both numerator and denominator are negative as x 3 - . 4. Evaluate the limit, if it exists (possibly as an inﬁnite limit). lim x 1 x 3 - 1 x 2 - 1 . lim x 1 x 3 - 1 x 2 - 1 = lim x 1 x 2 + x + 1 x + 1 = 3 2 . 5. Let f ( x ) = 3 + 1 /x . In the deﬁnition of the limit lim x →∞ f ( x ) = 3, if ε = 1 / 5, how large must

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Unformatted text preview: N be to guarantee that | f ( x )-3 | < ε for all x > N ? We need N ≥ 5. Then x > 5 gives | f ( x )-3 | = | 1 /x | = 1 / | x | < 1 / 5. 6. Show that the equation x 4-x-1 = 0 has at least one real solution in the interval (1 , 2). Let f ( x ) = x 4-x-1. Then f ( x ) is continuous, f (1) =-1, and f (2) = 13. Since-1 < < 13, there is at least one c ∈ (1 , 2) such that f ( c ) = 0, by the intermediate value theorem. 1 7. Diﬀerentiate the function f ( x ) = 2 x + 1 x + 3 . f ( x ) = ( x + 3) · 2-(2 x + 1) · 1 ( x + 3) 2 = 5 ( x + 3) 2 . 8. Diﬀerentiate the function f ( x ) = √ xe x . f ( x ) = 1 2 x-1 / 2 e x + x 1 / 2 e x = ( 1 2 x-1 / 2 + x 1 / 2 ) e x . 9. Find the values of x where the graph of y = x 3-6 x 2 has a horizontal tangent line. We must ﬁnd the values of x where y = 0. Now y = 3 x 2-12 x = 3 x ( x-4), so x = 0 or x = 4. 2...
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P MT 12 sln - N be to guarantee that | f x-3 |< ε for...

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