16A-FQ02

# 16A-FQ02 - at C(This question is an extra credit question(b...

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MAT 16A (A001) NAME: Final Exam Problem 1. ( estimated time: 15mn ) ( 20 points ) Find the derivatives of the following functions. 1 . f 1 ( x ) = x 3 - 2 x 2 + x - 1 2 . f 2 ( x ) = x 3 + 1 ( x - 1) 2 3 . f 3 ( x ) = p x 2 + x - 1 4 . f ( x ) = 3 2 3 p ( x 2 - x - 1) 2 5 . f 5 ( x ) = cos(2 x + 1) - x sin( - x + 1) Problem 2. ( estimated time: 50 mn ) ( 70 points ) Analyze and sketch the graph of the following function f ( x ) = x 2 - 6 x +12 x - 4 if x > 4 3 x 2 / 3 + 3 x 1 / 3 if x 4 . Problem 3. ( estimated time: 15 mn ) ( 20 points ) Sketch a graph of a continuous function f having the following characteristics. f ( - 2) = f (6) = 0 , lim x 7→-∞ f ( x ) = - 2 and lim x 7→ + f ( x ) = - 4 f 0 ( x ) > 0 if x < 2 , f 0 ( x ) < 0 if x > 2 , f 0 (2) undeﬁned and f 0 (4) = - 1 f 00 ( x ) < 0 if 2 < x < 4 , f 00 ( x ) > 0 if -∞ < x < 2 and 4 < x < Problem 4. ( estimated time: 30mn ) ( 30 points ) Consider a semicircle of radius r with diameter AB (see ﬁgure). (a) Show that for any point C in the semicircle, the triangle ABC is a right triangle

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Unformatted text preview: at C . (This question is an extra credit question ). (b) Find the dimensions of the right triangle of largest perimeter (with hypothenuse AB) that can be inscribed in a semicircle of radius r . 2 (c) Show that this right triangle has also the largest area among all right triangles (with hypothenuse AB) that can be inscribed in a semicircle of radius r . The problem 5 is a extra credit problem . Problem 5. ( estimated time: 10mn ) ( 20 points ) (a) Why Parabolas didn’t have inﬂection points? (b) Give an example of a continuous function whose graph crosses its horizontal asymp-tote....
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16A-FQ02 - at C(This question is an extra credit question(b...

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