**Unformatted text preview: **MTH405 Wk 9 Tutorial Explicit Dierentiation
Exercise. Dierentiate the following functions. 1.)f (x) = (4x2 − 1)(7x3 + x)
x3 + 2x2 − 1
3.)y =
x+5
2
5.)y = (x + 1)6 2.)f (x) = (x + 1)(2x − 1)
x2 − 1
4.)f (x) = 4
x +1
3
6.)y = (x + 2x)37 7.)y = [(x3 + 2x)2 + 13x]4
Exercise. Find the value of f 0 (3) of f (x) = (4x2 − 1)(7x3 + x). Exercise. Find the value of f 0 (4) of f (x) = (x2 + 1)6 Exercise. Dierentiate the following logarithmic and exponential func- tions. 3.)y = (ln x)3 x
3
3
4.)y = x ln x 5.)y = ln 2x3 6.)y = ex 7.)y = e5x 8.)y = e x 1.)f (x) = ln(x2 + 1) 9.)y = e(x−3x) 2.)y = ln 2 1 10.)y = cos(ex + 1) 1 Exercise. Dierentiate the following trigonometric functions. 1.)y = 10 sin x 2.)y = 3 cos 5t
5 − cos x
4.)f (x) =
5 + sin x
sin x sec x
6.)f (x) =
1 + x tan x
8.)y = sin x cos x 3.)y = 4x cos x + 2 sin x
sec x
1 + tan x
7.)f (x) = sin x2
5.)f (x) = Implicit Dierentiation
Exercise. following. Use implicit dierentiation to nd the derivative of the 1. x2 + y 2 = 100. 2. x2 y + 3xy 3 − x = 3. 3. x2 = 4. cos(xy 2 ) = y . 5. sin(x2 y 2 ) = x. x+y
x−y . + √1y = 1.
√
y = 3 2x − 5. 1
6. √
x
7. 2 Applications in Mechanics
1. An automobile is driven down a straight highway such that
after 0 ≤ t ≤ 12 seconds, it is
s(t) = 4.5t2 feet from its initial position.
(a)
(b)
2. Find the average velocity of the car over the interval [0, 12].
Find the instantaneous velocity of the car at t = 6. The distance x metres travelled by a vehicle in time t seconds
after the brakes are applied (deceleration of the vehicle) is given
by:
Determine
(a) (b)
3. 5
x(t) = 20t − t2 .
3 the speed of the vehicle (in km/h) at the instant the brakes
are applied, and
the distance the car travels before it stops. A missile red from ground level rises x metres vertically upwards in t second where
x(t) = 100t − 25 2
t.
2 Find
(a)
(b)
(c)
(d) the initial velocity of the missile,
the time when the height of the missile is a maximum,
the maximum height reached,
the velocity with which the missile strikes the ground. 3 Critical Points
1. For the function
f (x) = 24x + 6x2 − 4x3 , nd all critical numbers and determine whether each represents
a relative maximum, relative minimum, or neither. Then nd
the absolute extrema on the interval [−2, 3].
Determine the concavity of the function and identify any point
of inection of f (x).
2. For the function
f (x) = −2x3 − 3x2 + 12x, nd all critical numbers and determine whether each represents
a relative maximum, relative minimum, or neither. Then nd
the absolute extrema on the interval [−1, 2].
Determine the concavity of the function and identify any point
of inection of f (x). Optimization
1. 2. 3. An open rectangular box with square base is to be made from
60ft2 of material. What dimensions will result in a box with
the largest possible volume?
A rectangular plot of land is to be fenced in using two kinds of
fencing. Two opposite sides will use heavy duty fencing selling
at $3 a foot, while the remaining two sides will use standard
fencing selling for $2 a foot. What are the dimensions of the
rectangular plot of greatest area that can be fenced in at a cost
of $6000?
A rectangular area of 3 200 ft2 is to be fenced o. Two opposite
sides will use fencing costing $1 per foot and the remaining sides
will use fencing costing $2 per foot. Find the dimensions of the
rectangle of least cost. 4 ...

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- Spring '16
- Alveen Chand
- Velocity, Mathematical analysis, absolute extrema, following trigonometric functions, Implicit Di1Berentiation