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P MT 11

# P MT 11 - lim x → 3 √ x 1-2 x-3 5 Evaluate the limit if...

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Math 1A Calculus Fall, 2006 Prof. Haiman Practice Exam for Midterm 1 You may bring one sheet of notes to the exam (on ordinary sized paper, written on both sides). No other notes, books, calculators or computers may be used. This practice exam has more questions than the real exam. Show work or brieﬂy indicate a reason for each of your answers. For most questions, formulas showing one or two steps in your calculations, or a few words of explanation, will be suﬃcient. You can take the accompanying solutions to this practice exam as a guide to what is expected. 1. Find the domain and range of the function f ( x ) = 1 ( x - 2) 2 . 2. Express the function u ( t ) = cos t 1 + cos t as a composite f g of two other functions. 3. An exponential function f ( x ) = Ca x has f (1) = 10 and f (3) = 40. Find the constants C and a . 4. Evaluate the limit, if it exists (possibly as an inﬁnite limit).

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Unformatted text preview: lim x → 3 √ x + 1-2 x-3 . 5. Evaluate the limit, if it exists (possibly as an inﬁnite limit). lim x → π/ 2 tan x 6. Evaluate the limit, if it exists (possibly as an inﬁnite limit). lim x →∞ tan-1 ( x 2-x 4 ) 7. In the deﬁnition of the limit lim x → 1 (5-3 x ) = 2 , ﬁnd a value of δ that works for ε = 0 . 1. 8. Prove that there is at least one real solution of the equation xe x = 1. 1 9. For what value of the constant c is the function f ( x ) = ± x + c if x ≤ -1 x 2-c if x >-1 continuous on (-∞ , ∞ )? 10. Diﬀerentiate the function f ( x ) = ( √ x-1)( e x + x ) 11. Diﬀerentiate the function f ( x ) = 1 x 3-x + 2 12. Find an equation of the tangent line to the curve y = √ x at the point (4 , 2). 13. Diﬀerentiate e x (cos x + sin x ). 2...
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P MT 11 - lim x → 3 √ x 1-2 x-3 5 Evaluate the limit if...

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