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Calculus I Worksheet #77 1 A particle, initially at rest, moves along the x-axis so that its acceleration at any time t ≥ 0 is given by a(t) = 12t2 – 4. The position of the particle when t = 1 is x(1) = 3. (a) Find the values of t for which the particle is at rest. (b) Write an expression for the position x(t) of the particle at any time t ≥ 0. (c) Find the total distance traveled by the particle from t = 0 to t = 2. 2 Squares are to be cut out of the corners of a rectangular piece of cardboard and the sides folded up to make an open box. The rectangle is 10” x 5”. Find the side of the square to be cut out of each corner in order to maximize the volume of the box. 3 A rectangle is to be inscribed between the parabola f ( x) = 5 − x 2 and the x-axis, with its base on the xaxis. Find the value of x that maximizes the area of this rectangle. 4 A farmer has 555 feet of fence to enclose a rectangular field which is divided into six parts by his fence, as shown here: Calculate the overall maximum area that the farmer can fence off using his 555 feet of fencing. 5 The difference of a number x and its cube root 3 x is to be a minimum. Find the positive number that results in the minimum difference of the number and its cube root. 6 1 5 The position of a particle is given by x(t ) = t 4 − 2t 3 + t 2 + t + 2 . For what value of t, t ≥ 0 , is speed 4 2 greatest? The velocity of a particle is v(t ) = t 3 − 13t 2 + 47t − 30 . For what values of t, t ≥ 0 , is acceleration zero? 7 8 At which point on the following graph do both T S R P Q dy d2y and equal 0? dx dx 2 9 Find the minimum value of ƒ(x) = 7x2 – 14x + 12 10 If ƒ(x) = x4 – 4x3 a. Find intervals where ƒ is increasing b. Find inflection points of ƒ c. Find the absolute maximum value of ƒ on [–2,2] 11 Find c for Rolle’s Theorem for f ( x) = x 2 − 3 x − 4 on [-1, 4]. 12 Find dy if tan( xy ) = x 2 − 2 y dx Answers: 1) a. t = 0, 1 b. x(t ) = t 4 − 2t 2 + 4 c. 10 5) .192 9) 5 2) 1.057” 6) 3.528 10) a. ( 3, ∞ ) 3) x=1.291 4) 5500.466 ft 2 7) 2.569, 6..097 8) R 3 dy 2 x − y sec 2 ( xy ) 11) 12) = 2 dx 2 + x sec 2 ( xy ) b. (0, 0) & (2, -16) c. 48