MTH405_Wk_9_Tutorial - MTH405 Wk 9 Tutorial Explicit...

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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

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