Differentiation Questions and Answers

8 12 14 x 4 x 8 C log x 1 m 29 32 17 2 C 18 C 20 log e C 22 tan 7 4x C 4 26 2sin x C 1 m 24 log 2sin x 3cosx C 1 x 1 2 2 37 39 B logsin x C 9 x x 1 C C 15 log19 4x 1 C 16 34 2 tunx C cos ta cos 4 10 2log x 1 C tan x C 13 35 1 log x C NCERT 19 log e e 25 log 1 cos x C 31 1 18 2 3x 33 1 1 tin x 28 2V1 sinx C X tan 2x 3 x C sin x C 1 1 cos x 38 D C C C perda do log cosx sinx C 36 x log x C

The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells external hard drives The solid line represents the revenue function R x 40x the dashed line represents the cost function C x 5000 30x Use the formulas to find R 250 C 250 Describe what this means for the company R 250 C 250 Type an integer or a decimal 40 000 Ay 30 000 20 000 10 000 0 0 R x 40x C x 5000 30x X 200 400 600 800 Hard Drives Produced and Sold o

Determine whether 1 0 is a solution of the system 2x 5y 9 3x y 2 Is 1 0 a solution of the system Choose the correct answer below O Yes O No

11 tan 2x C 2 2 7 1 log x 1 C 8 log x x a C 9 log tanx tan x 4 C 10 log x 1 x 2x 2 C 6 13 log x 2 3x 2 C 15 logx 4 a b 2 16 2 2x x 3 C 18 log 3x 2x 1 20 x a x b C 6 log x 2 x 1 12 sin 9 19 6 9x 20 og x 2 x 9x 20 C C C 21 x 2x 3 log x 1 x 2x 3 C 22 log x 21 5 76108 1 6 1 2log x 1 C C X Rationalised 2023 24 23 5 x 4x 10 7log x 2 x 4x 10 C 24 B 25 B EXERCISE 7 5 1 log 3 log x 1 5log x 2 4log x 3 C 4 log x 1 2log x 2 log x 3 C 2 log 6 x go to he rept dished ANSWERS pe 443

Consider the four functions graphed in the figure below and assume the equations for A B C and D can all be written in the form y abt Which two functions have the same value for a B A click on image to enlarge D

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

Let y tan 2x 5 Find the differential dy when z 5 and dz 0 4 0 8574 Find the differential dy when z 5 and da 0 8 1 7148

Given the piecewise function defined by 3 x if x 1 H x x 1 if 1 x 3 x 3 if x 3 What is the value of H 3 H 3 OA 4 OB 6 OC 10 OR None of the above

hatch the functions with their graphs Enter the letter of the graph below which corresponds to the function Click on image for a larger view OD A OD 1 y 2 sin r 2 y 1 cos x 3 y cos 4 y 1 sin a NE B C D

Find the first derivative of 2x 1 4x In 2 f x tan

If f x 4 tan 6 sin 4x find f x 96 cos 4x 1 36 sin 4x

Which of the following functions are one to one Explain how you know a The function h graphed below b The function p graphed below

Part 1 of 9 a Find the formula for P t P t Type an expression using t as the variable Points 0 of 1 A sample of 12 grams of radioactive material is placed in a vault Let P t be the amount remaining after t years and let P t satisfy the differential equation P t 0 027P t Answer parts a through g

mework c How fast was the sample decaying after 6 hours d When was the sample decaying at the rate of 2 grams per hour a After 1 hour there were g remaining Round to the nearest integer as needed Part 1 of 4 A sample of radioactive material decays over time measured in hours with decay constant 2 The graph of the exponential function y P t shown on the right gives the number of grams remaining after t hours a How much was remaining after 1 hour b Approximate the half life of the material O C Points 0 of 1 975 3 1 P 4 3 5 7 Sav 9

Looking up Boris sees two hot air balloons in the sky as shown He determines that the lower hot air balloon is 515 meters away at an angle of 15 from the vertical The higher hot air balloon is 840 meters away at an angle of 22 from the vertical How much higher is the balloon on the right than the balloon on the left Do not round any intermediate computations Round your answer to the nearest tenth Note that the figure below is not drawn to scale 515 meters 15 22 840 meters meters X

Find x the angle of depression from the top of a lighthouse that is 165 ft above water level to the waterline of a ship 1143 ft off shore Round your answer to the nearest tenth of a degree 1143 U 165 0 x X

A ramp 22 ft long rises to a platform The bottom of the platform is 13 ft from the foot of the ramp Find x the angle of elevation of the ramp Round your answer to the nearest tenth of a degree Xx 13 Platform Ground X

A 12 ft ladder leans against the side of a house The top of the ladder is 10 ft off the ground Find x the angle of elevation of the ladder Round your answer to the nearest tenth of a degree 10 12 0 x

Evaluate S sinh 10x dx sinh 10x dx

The length of a shadow of a building is 56 ft when the sun is 64 above the horizon Find the height of the building Round your answer to the nearest tenth 000 56 X

Is the function y ex a solution of 3y 7y 4e X Choose the correct answer below Yes No Batt b Find 3y 7y and 3y 7y for y e x e 7 3 3y 7y 3y 7y

Find x Round your answer to the nearest tenth of a degree 15 A X

4 Suppose that f 4 2 g 4 5 f 4 6 and g 4 3 Find derivative h 4 in a d a h x 3f x 8g x b h x f x g x f x h x 1

1 4 7 10 12 14 6 440 cos2x 8 1 ax b 3a 1 3 3 16 x 3sinx e C 18 tan x sec x C 12 2 20 2 tan x 3 sec x 22 A 14 x C log x 2x C 4 cos cos x C 1 2 1 8 x C MATHEMATICS 11 3 x 4 x 8 C 2 log x m 1 m 8 1 17 C 1 2 2x 2 x 5 cos2x e 6 x C 6 5 sin 3x 3 EXERCISE 7 1 ax bx 2 3 1 3 4 x 1 x 1 C 4 NCERT cx C 9 logl x1 C 1 4a 3 7e e EXERCISE 7 2 not to be republished 17 3 2 x b C 7 x 2 x 2 C 3a 3 3x C 19 21 C cos 2 ax b C 18 etan x C Rationalised 2023 24 7 x 4 1 28 C 9 x x 1 C 10 2log x 1 C 5 4 3 tan x x C 13 5 3 log 1 log x C x 2 x C RT 10 x 3cos x x C 3 C 15 log19 4x 1 C 16 12 2 43 4 C 8 3 1 18 2 3x 19 log e e C kished

29 32 Find y and y 29 y x ln x 31 y In sec x In x 1 ln x 32 y In 1 ln x 30 y

2 26 Differentiate the function 2 g t In 3 1 3 f x ln x 3x 5 5 f x sin In x 7 f x In x 9 g x ln xe x 11 F t In t sin t 4 f x x ln x x 6 f x ln sin x 1 In x 8 y 10 g t 1 In t 12 p t In 1

3 5 11 log 15 3 2 2 tan 2x C 2 x x 4x 5 6 7 x 1 log x x 1 C 8 log x x a C 9 log tan x tan x 4 C log x 1 x 2x 2 C 13 log 22 tan 16 2 2x x 3 C 18 log 3x 2x 1 19 6 x 9x 20 Jong x 2 x 3x 2 C log x b x a x b C 2 C x 2 4 9 C 6 log 6 10 12 20 4x 4sin 2 c 21 2x 3 log x 1 1 2x 3 C 5 4log x 2 2log x 1 C 2 log x 25 5 log x 1 6 6 9x 20 C 15x 1 210g x 1 C 23 5 x 4x 10 7log x 2 x 4x 10 24 B 25 B EXERCISE 7 5 C 1 log 3 log x 1 5log x 2 4log x 3 C log x 1 2log x 2 log x 3 C Rationalised 2023 24 6 6 C C log x 3 C how to be rept lished ANSWERS log a log led

If f x 2 arctan 4x find f x Find f 4

Compute the following limit using l Hospital s rule if appropriate Enter Inf if your answer is o enter Inf if your answer is and ente lim 8 sin z ln z

If f x 5x arcsin z find f x 1 5 sin x Find f 0 8 272 3

Evaluate the limit using L Hopital s rule

If f x 4 tan 6 sin 4x find f x

If f x 5x arctan 7x find f x f x

Find the first derivative of f x f x 2 sin

Find the first derivative of f x tam

Anx Hindaun Road Bayana 17 August 20 The Principal Govt Senior Secondary School Bayana Subject Exemption from attending the evening games Reference Your Notice Dated 16 August 20 Sir I want to bring to your kind notice that my mother is not well She has been suffering from typhoid for two weeks I have to attend her in the evenings So I am unable to participate in evening games V Kindly exempt me from attending evening games for a period of fifteen days from today Hope you will oblige Regards

In the definition of limits we used the inequality 0 x x whereas in the definition of continuity we have used the inequality x x The reason is that in case of limits the function may or may not be defined at x but for the continuity of a function at the function must be defined on to Institute of Lifelong Learning University of Delhi lim f x f x X I Q 1 3 1 Continuity from the Right at a Point A function f X R is said to be right continuous or continuous form the right at a point ex if the right hand limit of f exists at x and equal to f x i e lim f x f x X X Continuity of Functions 3 2 Continuity from the Left at a Point A function f X R is said to be Left continuous or continuous form the Left at a point x X if the Left hand limit of f exists at x and equal to f x i e pg 4 4 Continuous Function A function f X R is said to be continuous on X if and only if f is continuous at each point of X Example 1 Show that the function f x is not continuous at x 0 Solution Given function is Moreover we know that f x since f x is not defined for x 0 therefore it cannot be continuous at x 0 Exfa 1 lim f x lim x 0 0X does not exist in R Hence f x is not continuous at x 0 Value Addition Note If the limit of a function f XR does not exist at a point x EX in R then the function f X R cannot be continuous at x Institute of Lifelong Learning University of Delhi pg 5

2 26 Differentiate the function 2 g t ln 3 1 3 f x ln x 3x 5 4 f x x lnx x

276 62X9 12

Let dy 0

Let f x 6 ln sec z tan z HINT Simplify the first derivative before you find the second derivative

Use logarithmic differentiation to find f x if f x f x 5x 3 3z 10

y

f x 96

Let f x In j f x 2x 3 9x 2 Then

Find the derivative of the function x n g x 3x x 2 e

et help logarithms f x 4 log

Let y ln x 4 Then dy da 11

Evaluate d In 7 at x 1

Let f x 27 In z Then x e