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Unformatted text preview: ESP
Kouba
Worksheet 1 O 1. Differentiate. a. y: 3x1 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
counterclockwise 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(5x7)
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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 Winter '09
 Kouba
 Calculus

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