espwksht10 - ESP Kouba Worksheet 1 O 1. Differentiate. a....

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Unformatted text preview: ESP Kouba Worksheet 1 O 1. Differentiate. a. y: 3x-1 7 b. f(x)=tan{$ln(3x+7)} tan 5 x 005 (7 x— 3) C- ' g(x) = sin (sin (sin (sin (1/x)))) d, y = c303 4142“ 2. Consider the function f(x) = X x2+1 a. Sketch its graph- b. What points (x, y) on the graph of f determine horizontal tangent lines ? c. What points (x, 'y) on the graph of f determine tangent lines with slope 1/2 ? d. What points (x, y) on the graph of f determine tangent lines with slope - 1/10 ? e. What is the smallest possible value of f' (x) ? 3. A lighthouse sits one mile offshore with a light beam turning counter-clockwise at the rate often revolutions per minute. See diagram. a. Write distance x asafunction of 6. b. Assume that both x and e are functions of time t- i. Determine deldt. ii. Determine dx/dt. 4. Gooh has a density of 200 grams per liter and sludge has a density of 250 grams per liter. What combination of gooh and sludge will result in 10 Titers of mixture (slooh 1’!) having a density of 238 grams per liter ? 5. Consider the diagram of nested circles A i and squares. The larger square has side length 2. Determine the area of the smaller circle. V v 6. Find all values of c guaranteed by the Mean Value Theorem for the following functions and intervals. a. f(x) x2—x on [0, 2] b. f(x) x1/3+1 on [2,0] 0. f(x) = x+ 1/x on [1/2, 3] d. f(x) = 5 sin3x on [0,1c] e. t(x) = cos(2x+1) on [0,1c/2] 7. Let d(t) = 10 t 2 + 3 J 2 t be the distance (miles) a bicyclist has traveled after t hours. Prove that, at least once during the first two hours of travel, the bicyclist reaches a speed of 23 miles per hour. 8. Let f and g be continuous functions with g(x)¢0 on the closed interval [a, b] and with f and g differentiable on the open interval (a, b) . Assume that f(a) = 9(a) and KM = 9(b)- Show that there is some number c in (a, b) satisfying 9(0) 1‘ ‘(c) = f(c) g ' (c) - 9. Determine a function fwhose derivative is given by f ' . a. f'(x)=3x2—-1 b. f'(x) = n + ‘17 c. f'(x) = 5 oos(5x-7) d. f'(x)=xse02x+tanx e‘ f'(x)= xcosx—sinx X2 10. Determineafunction y satisfying y‘ = 1.. 90’ ll. Prove that the equation x5 - \l x = x 1’3+1 has at least one solution c. ‘ . f 12.. the of = { 5m X Or x Z 1: 1r- x for x < 1t and prove that g is differentiable at x = K . l3. Sketch the graph of f(x) = x "5 and prove thatf is not differentiable mx=0. l ‘l. Tarzan can clean the bamboo hut in five hours. Jane can clean the hut in three hours. How long does it take the two of them working together to clean the hut ? 15’. a. Use your calculator to evaluate the first seven terms of the following sequence : a8 080$ 03 ,0808 ,o.8°B ,oaofl ,‘ Estimate what you think the limit of this sequence is- b. Let c be a mnstant so that the limit of the sequence c,c ,c ,‘c isz. Determine the value of c . ...
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espwksht10 - ESP Kouba Worksheet 1 O 1. Differentiate. a....

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