(1) Find the equations of the tangent and the normal lines
of 29/29
(1) Find the equations of the tangent and the normal lines to the graph of the given function at the indicated x value. (a) f (x )= sec x 4 at x = π () June 16, 2014 1 / 29
(1) Find the equations of the tangent and the normal lines
Text of (1) Find the equations of the tangent and the normal lines
(1) Find the equations of the tangent and the normal lines to the
graph of the given function at the indicated x value.
(a) f (x) = sec (x
4
(b) g(x) = x 3 √
x at x = 8
(2) Use the differentiation rules to find the first
derivative.
(a) y = 4x3− 1√ x
(b) f (t) = √
(c) r(θ) = sin θ tan2 θ
() June 16, 2014 6 / 29
(d) y = x + sin x x5 + 2x2
() June 16, 2014 7 / 29
(e) u = 4 √
(f) f (x) = cot (
(3) Find dy dx .
() June 16, 2014 10 / 29
(b) x2+y2 = √
(c) x2/3+y2/3 = 1
() June 16, 2014 12 / 29
(4) Find d2y dx2 in terms of x and y given xy + y2 = 2.
() June 16, 2014 13 / 29
() June 16, 2014 14 / 29
(5) Find the equation of the line tangent to the curve of x sin 2y
= y cos 2x at the point (π/4, π/2).
() June 16, 2014 15 / 29
(6) The volume of a cube is increasing at a rate of 1200 cm3/min at
the instant that its edges are 20 cm long? At what rate are the
lengths of the edges changing at that instant?
() June 16, 2014 16 / 29
() June 16, 2014 17 / 29
(7) A stone dropped into a still pond sends out a circular ripple
whose radius increases at a constant rate of 3 ft/sec. How fast is
the area enclosed by the ripple increasing at the end of 10
sec?
() June 16, 2014 18 / 29
() June 16, 2014 19 / 29
(8) Two commercial airplanes are flying at 40,000 ft along
straight-line courses that intersect at right angles. Plane A is
approaching the intersection point at a speed of 442 knots1 while
plane B is approaching the intersection point at 481 knots. At what
rate is the distance between the planes changing when A is 5 and B
is 12 nautical miles from the intersection?
1knots are nautical miles per hour and one nautical mile is 2000
yds () June 16, 2014 20 / 29
() June 16, 2014 21 / 29
() June 16, 2014 22 / 29
(9) A particle is moving along the x-axis so that its position s in
feet at time t in seconds satisfies
s = t4 − 8t3 + 10t2 − 4, t ≥ 0.
(a) Determine the average velocity over the interval [0,1]
() June 16, 2014 23 / 29
s = t4 − 8t3 + 10t2 − 4, t ≥ 0
(b) Find the velocity of the particle.
(c) Find the acceleration of the particle.
() June 16, 2014 24 / 29
s = t4 − 8t3 + 10t2 − 4, t ≥ 0 (d) Over which intervals is the
particle moving to the left, and over which is it moving to the
right?
() June 16, 2014 25 / 29
s = t4 − 8t3 + 10t2 − 4, t ≥ 0
(e) At which times is the particle at rest?
() June 16, 2014 26 / 29
s = t4 − 8t3 + 10t2 − 4, t ≥ 0
Argue, with some solid mathematics, that at some moment between t =
0 and t = 7 sec, the particle must be at the origin.
() June 16, 2014 27 / 29
() June 16, 2014 28 / 29
() June 16, 2014 29 / 29