Download Calculus I Worksheet #77 1 A particle, initially at rest, moves along

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