Calculus Questions

Check whether each of the following functions is a solution of the differential equation 3y 7y 4e x X b y e e 7 3 x a y ex c y ex Ce 7 3 x a Find 3y 7y and 3y 7y for y ex 3y 7y 3y 7y

Find x Round your answer to the nearest tenth of a degree 10 14 x x

Find dy dt y cos nt 13

0 13 Points DETAILS A graphing calculator is recommended a How is the number e defined SCALCET9 Oe is the number such that lim h 0 Oe is the number such that lim h 1 O e is the number such that lim h00 lim h 0 lim h 0 eh 1 1 h h e O e is the number such that lime h 0 O e is the number such that lim 1 1 h eh 1 1 h e h 1 1 h b Use a calculator to estimate the values of the limits correct to two decimal place 2 7h 1 h 2 8h 1 h 1 1 h What can you conclude about the value of e O 0 99 e 1 03 O 0 99 e 2 7 O 1 03 e 2 7 O 2 7 e 2 8 0 0 e 1

63 78 Find the derivative of the function Simplify where possible 63 f x sin 5x 65 y tan x 1 64 g x sec e 66 y tan x

larger x value 13 0 13 Points Draw a diagram to show that there are two tangent lines to the parabola of y x that pass through the point 0 16 Find the coordinates of the points where these tangent lines intersect the parabola smaller x value x y Need Help Submit Answer DETAILS 14 0 13 Points x y Read It SCALCET9 3 1 067 DETAILS Watch It SCALCET9 3 1 074 Find a parabola with equation y ax bx c that has slope 14 at x 1 slope 18 at x 1 and passes through the point 2 37

Find an equation for the plane that is tangent to the surface z In x y at the point P 0 1 0 Using a coefficient of 1 for z the equation of the plane is 0

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 Evaluate the limit if it exists otherwise give the reason why the limit does not exist a lim cos 2 N 1 b lim 4x x 2x

a Find the velocity and acceleration as v t a t b Find the acceleration in m s2 after 9 seconds m s c Find the acceleration in m s2 when the velocity is 0 m s Need Help Submit Answer 1 0 13 Points Read It DETAILS Watch It ww SCALCET9 3 1 059 Find the points on the curve y x 3x 9x 1 where the tangent is horizontal

12 0 13 Points DETAILS SCALCET9 3 1 064 O At what exact point on the curve y 7 2e 5x is the tangent line parallel to the line 5x y 8 x y Illustrate by graphing the curve and both lines y 4 3 2 1 Need Help Read It 15 10 5 SA 1 2 3 X O 3 2 1 15 x S 5 1 2 3 4 X 3 2 1 15 5 5 1 2 3 4 X 4 3 2 1 1

1 log 1 x C 4 cos cos x C 440 8 12 11 x 4 x 8 C 3 14 17 2 6 ax b C 7 x 2 x 2 C 3a 20 22 MATHEMATICS 32 log x l m 1 m 37 1 2e 1 2x C 9 x x 1 C 9 C 26 2sin x C C log e x e x C 1 29 log sin x C x 2 x 14 x 1 C tan 7 4x C 4 39 B 2 5 1 2 24 log 2 sin x 3 cos x C 15 4 3 logl x C cos tan x C 4 4a cos 2 ax b C Rationalised 2023 24 18 etan x C 8 3 34 2 tan x C 35 1 log x C 3 log 1 log x C 10 2log x 1 C e2x 3 log19 4x 1 C 16 3 C NCERT 13 Rationalised 2023 24 19 log e log 1 cos x C 31 25 1 18 2 3x 33 1 1 tan x 28 2 1 sinx C X tan 2x 3 x C 1 sin x C 1 1 cos x 38 D 1 C 22 C C otto e republished log cos x sinx C 36 x log x C 3

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

10 g t 1 In t

Evaluate the limit using L Hopital s rule lim 3 cos 52 sec 7x

a If the symbol x denotes the greates integer function defined in this example evaluate the following If an answer does not exist enter DNE i lim 12 ii lim x 2112 iii lim ain x b If n is an integer evaluate the following i lim ii lim 2 3 2 n c For what values of a does lim exist x The limit exists only for a 0 The limit exists for all non integer values of a The limit exists for all positive values of a

Compute the following limits using l Hospital s rule if appropriate Enter Inf if your answer is enter Inf if your answer is and en 32 3 z122 1 tan 2 lim zxx 1 2 3 11

Use continuity to evaluate the limit 40 9 40 x Consider the intervals for which the numerator and the denominator are continuous The numerator 40 is contintuous on the interval lim The denominator 40 is continuous and nonzero on the interval The interval for which the quotient is continuous is the intersection of the above intervals 40 Therefore the quotient 40 x Since 9 is in the interval 0 then f is continuous at x 9 Therefore 40 lim 1 9 40 x is continuous on the interval f 9

Find the first derivative of f x f x r tan

Find the domain of the function I 4x 3x 1 X g x 0 0 00 1 4 U 1 4 O 1 U 4 00 O 0 1 4 U 1 4 00 0 4 U 4 00

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

If f x sin 5x 2 then f x Note The inverse of sin z can be entered as arcsin r or asin

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

Evaluate the following limit Enter Inf if your answer is 2x ta lim z 0 5x ta enter Inf if your answer is oo and enter D

Show that the following limit is true 1 lim 240 First note that we cannot use 1 lim sin lim 2 0 I 2 0 sin I sin 1 lim sin I 0 1 because the limit as a approaches 0 of sin does not exist see this example Instead we a I 1 the Squeeze Theorem and so we need to find a function f smaller than g x x sin function h bigger than g such that both f x and h x approach 0 To do this we use our knowledge of the sine function Because the sine of any number lies between and we can write x 1 x and Any inequality remains true when multiplied by a positive number We know that a 0 for all a and so multiplying each side of inequalities of x we get 1

Evaluate the limit using L Hospital s rule if necessary 7100

Evaluate the limit using L Hospital s rule sin 15 lim tan 5

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

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

Evaluate the limit using L Hospital s rule if necessary et lim 70 sin 4x

Find the first derivative of f x f x 2 sin

Find the first derivative of f x tam

a What is wrong with the following equation 2 x 42 I 6 O x 6 x 7 x x 42 O The left hand side is not defined for x O The left hand side is not defined for x O None if these the equation is correct x 7 b In view of part a explain why the equation x x 42 x 6 lim 2 6 is correct Since x x 42 x 6 lim x 7 2 6 0 but the right hand side is 6 but the right hand side is and 7 are both continuous the equation follows O Since the equation holds for all x6 it follows that both sides of the equation approach th same limit as a 6 This equation follows from the fact that the equation in part a is correct None of these the equation is not correct

Find the limit if it exists If an answer does not exist enter DNE 10x 50 x 5 lim 2 5

Evaluate the limit below 2 h 4 h lim h 0 If we define F h then we can t compute the limit by letting h 0 since F 0 is undefined But if we simplify E algebraically we find that F h lim h 0 2 h 4 h Recall that we consider only h 0 when letting h approach 0 Thus 2 h 4 h lim h 0 Question Help Message instructor A

lim f x 9 x 1 b lim g x x 1 find the limits if they exist If an answer does not exist enter DNE a lim f x 2g x x 1 c lim 2 1 4f x x 1 g x d lim f x g x 2 1 h x e lim f lim lim g x 4 2 1 g x h x f x lim h x 0 2 1

Evaluate the limit using the appropriate Limit Law s If an answer does not exist enter DNE t4 5 lim t 2 2t 3t 9 11 19 X Question Help Message instructor

Find the limit if it exists 6x 30 lim 2 5 x 5 Recall that x 5 x 5 if x 5 x 5 if x 5 Therefore we will need to check the limits when approaching from the left and from the right We will start by checking the limit when approaching from the left As a approaches 5 from left we have lim 2 5 6x 30 x 5 Next we check the limit when approaching from the right As a approaches 5 from the right w have 6x 30 lim 5 x 5 Since the left and right limits are Select an answer exist enter DNE 6x 30 lim 2 5 x 5 Question Help Video M the limit is as follows If an answer does not

Evaluate the limit if it exists If an answer does not exist enter DNE x 6x 9 24 81 lim 24 3

Find the limit If an answer does not exist enter DNE sin 8 lim 0 0 70 tan 0

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

Name Instructions Use algebraic techniques to solve problems unless otherwise instructed Show the calculations leading to your answers except basic algebra arithmetic when there is work to show Simplify and circle your final answers 1 18 pts In the vicinity of a bonfire the temperature T in C at a distance x meters from the center of the fire is described by T x 500 000 x 400 a What is the average rate of change of T between 0 and 10 meters Also what is the interpretation of this average rate of change Use correct units in your interpretation b What is the average rate of change of T between 10 and 20 meters Also what is the interpretation of this average rate of change Use correct units in your interpretation c What is the equation of the secant line through 10 T 10 and 20 T 20 Write the answer in slope

Ans 115 Civil Lines Ajmer 25 July 20 The Principal Dayanand Senior Secondary School Ajmer Subject To Issue Transfer Certificate Sir I am a student of class XI C of your school My father is a clerk in the State Bank of India Ajmer But now he has been transferred to Suratgarh I have to secure admission to a school at Suratgarh You are therefore requested to issue me my T C No dues certificate is attached herewith Regards Yours obediently Nand Lal S o Mr Hira Lal S R No 326 20 19 A Letter for Railway Concession