021.11s.midterm

021.11s.midterm - ´(b Determine all(vertical and...

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HKUST MATH021 Concise Calculus Midterm Examination Name: 6 Apr 2011 Student I.D.: 12:30–13:20 Signature: Directions: Do NOT open the exam until instructed to do so. All mobile phones and pagers should be switched OFF during the examination. You must show the steps in order to receive full credits. Electronic calculators are NOT allowed. This is a closed book examination. Answer ALL questions. Some formula: sin( x + y ) = sin x cos y + cos x sin y sin( x - y ) = sin x cos y - cos x sin y cos( x + y ) = cos x cos y - sin x sin y cos( x - y ) = cos x cos y + sin x sin y tan( x + y ) = tan x +tan y 1 tan x tan y sin 2 x = 2 sin x cos x cos 2 x = cos 2 x - sin 2 x = 2 cos 2 x - 1 = 1 - 2 sin 2 x sec 2 x = tan 2 x + 1 Question No. Points Out of 1 6 2 8 3 6 4 10 Total 30 1

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Answer all questions. Show all your work for full credit. 1. Compute the following limits. (a) lim θ →∞ sin 2 πθ θ = . (b) lim x 0 + x x = . 2
2. Let f ( x ) = e x + 1. (a) The range of f is . (b) Given f is one-to-one on its domain. Compute the inverse of f . (c) The anti-derivative of f is . 3. Compute the derivative of f ( x ) = - x 2 using the definition of differentiation . 3

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Unformatted text preview: ´ (b) Determine all (vertical and horizontal) asymptote(s) of f ( x ). (c) Find f ′ ( x ) and f ′′ ( x ). f ′ ( x ) = ± ² ³ ´ and f ′′ ( x ) = ± ² ³ ´ . (d) Write down all critical point(s) of f ( x ): ± ² ³ ´ . (e) Find the interval(s) on which f is increasing and also the interval(s) on which f is decreasing. f ( x ) is increasing on the interval(s) ± ² ³ ´ and f ( x ) is decreasing on the interval(s) ± ² ³ ´ . 4 (f) Find the interval(s) on which f is concave up and also the interval(s) on which f is concave down. f ( x ) is concave up on the interval(s) ± ² ³ ´ and f ( x ) is concave down on the interval(s) ± ² ³ ´ . (g) Write down all inﬂection point(s) of f ( x ): ± ² ³ ´ . (h) Write down all root(s) of f (all x such that f ( x ) = 0): ± ² ³ ´ (i) Sketch f ( x ). 5- This page intentionally left blank for scratch work -6...
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