Differentiation Questions and Answers

Let y ln 3t 11t 7 Then dy

Estimate the value of lim 1 0 x 4 21 lim 21 24 Use a table of values to estimate the value of the limit If you have a graphing device use it to confirm your result graphically x 1 23 lim x 1 X 1 10 1 9 t 2 lan 3x x 0 tan 5x 22 lim U 24 lim x 0 3 9 5 X

Consider the contour diagram for the function z f x y below 2 1 2 0 Z 0 1 O 1 only O II only O III only O 1 and II II and III Z 2 2 Au P 3 V Z 4 4 O 5 Z 6 6 X Based on the contour diagram which of the following statements are true 1 The directional derivative of fat Pin the direction of u is positive II The directional derivative of fat Pin the direction of v is positive III The directional derivative of fat Qin the direction of w is negative

roblem 7 The following limit represents the derivative of some function fat some number a Find such an fand a lim h 0 25 h 5 h A x x a 25 h B f x x a 25 C f x x a 5 D f x x h 5 a E f x x h a 25 F f x x h a 5 G f x a 5th ann

23 If f x x x 10 and f 1 2 find f 1

Given f x x 4 x 3 x 2 a f 0

16 sin xy cos x y

12 esin y x y 14 tan x y 2xy 1 16 sin xy cos x y

14 tan x y 2xy 1

X 3 2 16 pts Consider f x 5 X 6 and g x X 4 the results if possible and describe their domains a f g x Find the rules of the new functions below simplify

10 xe x y 12 e sin y x y

12 e sin y x y

5 22 Find dy dx by implicit differentiation 5 x 4xy y 4 7 x x y 5 6 2x xy y 2 8 x xy y 1

53 56 Find y and y 53 y cos sin 30

Find by implicit differentiation dy dx dy dx 5 sin x cos y sin x cos y

S No Vechile No 1 RJ29PA1284 CHC Geejgarh RJ29PA1283 PHC Rahuwas RJ29PA1286 CHC Paparda RI29PA1289 PHC Bhandarage 2 3 4 5 6 7 00 8 RJ14PE6665 11 RI29PA1290 PHC Lotwara RJ14PE6667 Location R 29PA1285 PHC Guda Katla CHC Lalsot 9 RJ29PA1287 PHC Santha RI29PA1742 10 RJ29PA1288 PHC Khedla 12 RJ29PA1744 CHC Mahwa CHC Manpur Sikrai CHC Mandawar 13 R129PA1720 PHC Sainthal Morden Emergency Service Privious ODO Meter 11042 340103 10246 203393 62075 17039 255256 48765 Pilot Name zibar RRARDOT Ayol RIGH VEGH disay Sakata GUES E Current ODO Meter 110 50 11334 131812 219441 340105 10251 203399 62177 77050 2 55338 48831 171782 171782 5272 5278 11336 13 8 2 219441 Case 1 1 1 1 6 1 5 3 1 ODO Meter Difference 8 2 S 6 102 77 66 0 6 2 0 O Date 1 9 2023 GPS KM 7 O S O 102 17 83 64 LD 1 0 GPS Total Km 21557 0 25067 41 23171 56 O 17210 92 59770 27769 5 50647 46995 34272 2540 s 15931 380 25

1 04 Points NOTES Find y and y y ASK YOUR TE DETAILS y e e3e

Let x 2x cos 2x f x 4x cos 2x 4x sin 2x f 5 20 cos 10 100 sin 10 f x 4 cos 2x 16x sin 2x 18x cos 2 5 80 sin 10 196 cos 10

2 Let y 4x 7x 4 Find the differential dy when 4 and dx 0 2 Find the differential dy when x 4 and da Sam 0 4

If z 8 and dy dt 5 when 2 and y Zensk dy dt 2 what is when 2 and y 2

find x cos x 8x sin x 4x Find 2 4x tan x 4x 2 tan x 2 sec x f x 42 ta sec

Now let a be any arbitrary element of R then there arise two cases Case I If a 0 then and 1 lim f x lim x sin X NI 2 M X Now f a asin 1 asin 1 a whenever Continuity of Functions a Then lim f x f a a sin X M SITY I Q 2 I Q 3 1 Thus f x is continuous at x a Case II If a 0 then by the definition of function f a f 0 0 a x 0 xsin 0 si S A Sin JO f x f 0 Now if we choose s then for a given 0 there exists a number 8 E 0 such that That f x f 0 e x 0 x 8 Thus f x is continuous at x 0 Hence f x is continuous on R OF DEL Institute of Lifelong Learning University of Delhi Continuity of Functions pg 8 8

Find y by implicit differentiation Match the equations defining y implicitly with the letters labeling the expressions for y 1 5x cos y 4 cos 2y 3 sin y 2 5z cos y 4 sin 2y 3 sin y 3 5z sin y 4 cos 2y 3 cos y 4 52 sin y 4 sin 2y 3 cos y A y B y C y D y 5x cos y 5x sin y 5 sin y 8 cos 2y 5 cos y 3 sin y 8 cos 2y 3 cos y 5 cos y 5x sin y 8 sin 2y 3 cos y 5 sin y 8 sin 2y 5x cos y 3 sin y

Sammen A particle is moving along the curve y 3 4z 1 As the particle passes through the point 2 9 its coordinate increases at a rate of units per second Find the rate of change of the distance from the particle to the origin at this instant

The radius of a spherical balloon is increasing at a rate of 4 centimeters per minute How fast is the volume changing in cubic centimeters per minute when the radius is 14 centimeters Note The volume of a sphere is given by V 4 3 Tr Rate of change of volume in cubic centimeters per minute

Find the slope of the tangent line to the curve at the point 77 7 2 4 sin z 3 cos y 2 sin r cos y z 7

3 Let y 28 Find y x at the point 3 1 3 3

Find dy de by implicit differentation dy da 4 2x sin xy

dy If y xy 3x 7 and dr 2 when a 3 and y Genti 1 what is when 3 and y 1

If z y 8 and dy WI dr dt 5 when t 2 and y 2 what is dy when 2 and y 2

For the equation given below evaluate y at the point 1 2 y at 1 2 5x y 2y 97

Let f x 2x 3 2 2 x 3 4 2x 3 2x 3 5

For each of the following integrals give an appropriate trigonometric substitution of the form f t to simplify the integral x 9 x dx a 2 9 x 49 dx b C d 5x2 8 dx 6 dr H 11 x C

Plot the ordered pair 7 6 on the rectangular coordinate plane Plot 7 6 a 2 2 10 B 6 10 8 84 2 Ay 4 6 8

3 a What does the limit lim 0 f h 4 f 4 h 8 indicate about the function f x x

1 27 100 1 000 1 4 81 x 33 0 25 2 4 36 5 7x 6 30

1 36 32 0 01 2 100 5 29 62 x 3 35 4 7 15 38 3x 2 36

5 5 x 9 8 b 5 3 2 1 11 5x 40 4 14 2x 3 17 z 81

13 14 4b 2

3 x 4 X3 6 b 8 9 2y 12 1 12 5x 7 22 7 81 215 3a

4 15 x y6 x y5

3 2x 6 a What is the domain of the function 3 Let y b What is the range of the function

3 PSEUDO GRAPH Definition If an object contains multiple lines and loops then it is called a pseudo graph Fig 1 7 is a pseudo graph Fig 1 7 Note If G is a p q graph then q p C and q p C if and only if any two distinct points are disjoint 4 COMPLETE GRAPH Definition A graph in which any two distinct points are adjacent is called a complete graph A complete graph with p vertices is denoted by K p K K 0 K K Fig 1 8 Note The number of edges of a complete granh K is n C Ks

7 52 Find the derivative of the function 7 f x 2x 5x 4 5 8 f x 9 f x 5x 1 10 f x 11 g t 1 2t 1 13 f 0 cos 0 15 g x ex x 12 F t 14 g 0 16 y 5 18 f t 20 A r 17 y x e 3x I 19 f t et sin bt 21 F x 4x 5 x 2x 5 4

Let f x xlxl a Sketch the graph of the function y 2 V 4 2 2 2 2 c Find a formula for f f x y ALCE19 2 8 061 2 b For what values of x is f differentiable Oxz0 Ox 0 Ox 0 0 all real x 2 X 4 X 4 4 2 2 y 4 N 2 T y 2 2 2 X X

f x b Check to see that your answer to part a is reasonable by comparing the graphs of f and f A graphing calculator is recommended 4 f 4 2 2 y 100 50 50 100 y 100 50 50 100 2 4 4 f X S 4 4 2 2 y 100 50 50 100 y 100 50 50 100 2 2 4 4 X X

Problem 10 Let A be a 3 10 matrix and let B be m n Suppose that BTAT is a 21 x 3 matrix What is the value of m Problem 10

Problem 8 Let A and B be square matrces of the same size Which of the following statements are always true i tr 47 tr 4 ii tr cA c tr 4 where c is a scalar iii tr A B tr 4 tr B A ii and iii only B i and ii only C iii only D all of them E ii only F 1 only G none of them H i and iii only

Problem 6 Suppose that you were to try to find a parabola y ax bx c that passes through the x y pairs 4 15 3 2 and 2 1 To obtain the coefficients a b and c you would try to solve a system of linear equations whose augmented matrix is which of the following Problem 6 A 225 4 1 4 3 1 1 2 1 16 E 9 4 Select v 4 1 3 1 2 1 15 2 1 225 4 1 1 16 15 2 1 3 1 1 2 B 9 4 F 225 4 1 15 1 16 2 1 9 1 1 4 16 C 9 4 4 1 15 1 2 3 2 1 1 D 225 4 15 1 4 2 1 3 1 1 2

A cup of coffee has its temperature F in degrees Fahrenheit at time t given by the function F t 80 95 e 0 05t where time is measured in minutes Use central difference with h 0 01 to estimate the value of F 10 Round to the nearest tenth 2 9 What are the units on the value of F 10 degF per min What is the practical meaning of the value of F 10 The coffee has decreased by 10 degrees over 2 9 minutes The coffee s temperature is decreasing at about 2 9 degrees per minute The coffee s temperature has decreased by about 2 9 degrees over ten minutes Part 1 of 2 Which do you expect to be greater F 10 or F 20 F 10 S Write a sentence that describes the behavior of the function y How do you think its graph will look Why Part 2 of 2 F t on the time interval 0 t 30