Differentiation Questions and Answers

1 22 Differentiate 1 f x 3 sin x 2 cos x 3 y x cotx 5 h 0 0 sin 0 7 y sec 0 tan 0

Evaluate the double integral of the function over the rectangle Y 1dA R 0 28 x 2 1

Use formulas to find the area of the figure 3 mi b 7 mi 2 OA 12 mi 2 OB 27 mi OC 24 mi OD 10 5 mi 18 mi 8 mi

Simplify the expression to a bi form 4 175 100 112

Find the average rate of change on the interval specified for real numbers b where b 3 g x 3 2 4 on 3 6 Basic Funcs Trig X rum

Find the area of the figure A Simplify your answer 12 m 26 m 48 m 26 m 12 m 22 m

1 Determine if each of the following relationships can be represented with a function f where f z y Give a reason for your answer a A child s height in inches y on their birthday at age z b month of the year y average weekly price of gas at a gas station c The cost of mailing a small package y based on its weight z

2 Let h x x 3x 6 Find the following values a h 3 b h 2

Get a similar question You can retry Suppose f x is continuous over an interval a x b and a c d b 10 then check all that apply f x is decreasing through a c f does not have a maximum at x c Of does not have a minimum at x c 10 Of c f c exists f x is increasing through x c X Hint Be sure to check ALL that apply X If f d 1 then check all that apply f does not have a maximum at x d f x is increasing through x d f d 1 Of x is decreasing through x d Of does not have a minimum at x d Of d exists Hint Be sure to check ALL that apply Based on this we know that for some x between x cand x d f does not exist Of has a minimum Of has a maximum f x 0

5 4 3 2 1 B 3 2 t N 3 23 A Determine the intervals where the derivative of this graph is positive and negative Use interval notation Positive derivative on the interval s Negative derivative on the interval s

Question 10 The point 15 3 lies on a differentiable function f x The tangent line to f x at x 15 also goes through the point 2 88 What is f 15 f 15

15 54 to find the indicated derivative 1 f 1 if f x x 3 f 3 if f t 8 1 Microsoft Teams 4 In Problems 5 22 use f x lim f x h f x h to find the derivative at x https teams microsoft com modern calling 5 s x 2x 1 7 r x 3x 4 9 f x ax bx c 11 f x x 2x 1 13 h x Latin Dals Edin Powell and tege Wed 14 132 X f c lim AD 100 9 y sin x x 11 y cos x 13 y 19 y 3x x 2 15 y cos 17 y 3x 2 3 x x 1 3x 4 24 10 12 y sin 3x 14 y 3 16 y cos 18 y 2 3x x 3 20 y 24 D Suara Anda dimatikan Tekan Ctrl Shift M untuk menyalakan suara mikrofon atau tekan dan tahan Ctrl Spacebar 2 f 2 if f t 2x 4 f 4 if f s 2x 3 x 4 In Problems 21 28 find the indicated derivative 21 y where y x 4 22 y where y x sin x Orang Angkat 6 f x ax 8 f x x x 1 10 f x x 12 g x x x 14 S x 88 Reaksi Tampilan Lainnya 39 Comment sin x 3x Sin cost 37 D cos sin 0 A Share Kamera Mikr sin cos sin 2x 4 In Problems 41 46 use Figures 2 a cated expressions CO 3

3 30 Differentiate 3 y 4x2 3 2x 5 I 4 y 10x 7x 2 2x 5 y x e 7 f x 3x 5x e

Complete the table X 2 1 0 1 2 Sketch a graph of f f x x 3 720 x y

In Problems 1 4 use the definition f c lim AD f c h f c h to find the indicated derivative 1 f 1 if f x x 3 f 3 if f 1 1 5 s x 2x 1 7 r x 3x 4 9 f x ax bx c 11 f x x 2x 1 2 13 h x X 2 f 2 if f t 2x 4 f 4 if f s In Problems 5 22 use f x lim f x h f x h to find the derivative at x 6 f x ax 8 f x x x 1 10 f x x 12 g x x 1 14 S x

In Problems 1 4 use the definition f c lim 0 to find the indicated derivative 1 f 1 if f x x 3 f 3 if f 1 2 1 In Problems 5 22 use f x the derivative at x 5 s x 2x 1 7 r x 3x 4 9 f x ax bx c 11 f x x 2x 1 13 h x IN f c h f c h X 2 f 2 if f t 2x 4 f 4 if f s lim x h f x h to find 6 f x ax B 8 f x x x 1 10 f x x 12 g x x x 14 S x X 1

Find the domain of the functions 1 f x x 1 2 f x 5 x x 3 5 4 0 0 5 4 10 8 15 12

For any continuous piecewise linear function F on the interval a b we have x so F b F a x b a What is the relationship between x and the slopes of the lines on F on the interval a b Describe with examples and great detail

Use the graph of the function f x below to find the average rate of change over the interval 6 1 Note that you can drag the point anywhere along the curve Average Rate of Change m 10 30 20 10 1 5 6 25 0 10

41 42 Find equations of the tangent line and normal line to the curve at the given point 41 y x 2e 0 2 42 y x 2 1 1

ROME WAST OVERNAN ARRAY THEATER 20 2 2 WHISTE DAE 1950 17 g x DONE 1 x x 19 f x x x 3 4 x 21 y 3e

5 Suppose lim f x 1 lim f x 1 and f 3 is not defined Which if any of the following Statements may be false A lim f x 1 3113 B f is continuous everywhere except at x 3 C If we redefine f 3 to be equal to 1 then the new function will be continuous at x 3 D f has a removable discontinuity at x 3 E All of the preceding statements are true

Solve the initial value problem d s dt 162 cos 9t s 0 13 s 0 8 OA s 18 cos 9t 13t 10 OB S 18 sin 9t 13 OC s 2 cos 9t 13t 6 OD s 2 cos 9t 6

Use the English and metric equivalents to the right along with dimensional analysis to convert the given measurement to the unit indicated 20 m to yd yd Round to the nearest hundredth Approximate English and Metric Equiv 2 54 cm 30 48 cm 1 in 1 ft 1 yd 1 mi 0 9 m 1 6 km

60 ft A GOOOD A slow pitch softball diamond is actually a square 60 ft on a side He from home to second base It is Round from home to second base tenth as needed ft in yd

Solve the initial value problem d s dt 3 2 7 81 sin 9t OA s sin 9t 91 OB s sin 9t n st 3 2 OC s sin 9t 3 2 O D s 9cos 9t s 0 8 s 0 2 7 3 2 3 2 8t 3 7t 2 8

Simplify by taking roots of the numerator and the denominator 3 3 729 8 729 8 Simplify your answer Type an exact answer using radicals as needed

Divide Then simplify by taking roots if possible Assume that all expressions under radicals represent positive numbe 40521 25 5x 7 17 21 25 405x y

ply 5 21 91 5 2 0 Type an exact answer us

Multiply and simplify by factoring 27 24 The answer is Type Ints 0 an exact answer

Multiply and then simplify by factoring if possible 2 10 2 10 Type an exact answer using radicals as needed

Simplify by factoring Assume that no radicands were formed by raising negative numbers to even powers 3 270 270 Type an exact answer using radicals as needed

Find the value of S Determine whether N 4dc x 1 4 n 1 Enter C if series is convergent D is a convergent series if series is divergent

From the graph of the function state the intervals on which the function is increasing decreasing or constant The function is increasing on what interval OA 1 00 OB 1 00 C 3 00 OD 3 00 The function is constant on what interval OA 3 3 OB 3 3 OC 1 6 OD 1 3 Clear all x Skil

Determine any relative maxima or minima of the function and the intervals on which the function is increasing or decreasing 2 f x x 9x 17 Does the function have a relative maximum or minimum O Relative minimum O Relative maximum 10 10 10 Ay 4 5 3 25 10 C 6 C

For each of the following series indicate whether the integral test can be used to determine its convergence or not and if not why cos n A M Can the integral test be used to test convergence O A no because the terms in the series do not decrease in magnitude B no because the terms in the series are not all positive for n c for some c 0 C no because the series is not a geometric series D no because the terms in the series are not recursively defined O E no because the terms in the series are not defined for all n O F yes B n 7 Can the integral test be used to test convergence A no because the terms in the series do not decrease in magnitude B no because the terms in the series are not all positive for n c for some c 0 C no because the series is not a geometric series D no because the terms in the series are not recursively defined E no because the terms in the series are not defined for all n F yes

2 H 10 9 8 6 5 4 3 a Give the intervals where the function is continuous Answers are expected in interval notation e g 00 2 U 2 00 Use U between intervals and use oc oo for negative and positive infinity and are differentiated Unless explicitly shown in the image ass the left and right endpoints are finite and closed

11 Describe the parallels between finding the instantaneous velocity of an object at a point in time and finding the slope of the line tangent to the graph of a function at a point on the graph

x Cos y Use derivatives and exact partial derivatives at 1 56 2 1 Use the finite difference formulas to estimate af and off at 1 56 2 1 for three different x dy values of the mesh size differentiation rules to find the exact partial af f x and evaluate those o h 0 01 o h 0 001 o h 0 0001 dy 9

Consider the function f x y cos 9y Find and classify all critical points of the function If there are more blanks than critical points leave the remaining entries blank f fay 4 1

Make a table of values and sketch the graph of the equation Find the x and y intercepts and a x 3y 6 Make a table of values and sketch the graph of the equation y 10 10 5 y intercept x y 5 10 5 5 10 y 10 5 5 10 5 Test for symmetry Select all that annly 5 10 10 X Find the x and y intercepts If an answer does exist enter DNE X intercept x y X O 11 10 5

If f x 0 10z sin determine whether or not f 0 exists 0 Answer Yes or No 2 if x if x

The limit f x lim hs0 0 2 h h is the standard expression defining the derivative of some function f x at some number a Find and a 4

2 12 pts For f x x 3 a calculate f 5x and 5f x and b f x 2 and f x f 2

Labie gives the number w Q d t 2000 2002 2004 t 2006 2008 2010 2012 2014 N t thousands 2000 306 5 2002 900 2004 497 5 2006 899 25 5 520 2008 603 25 2010 892 4 877 7 450 What are its units O thousands of surgeries years thousands of surgeries per year years per thousand surgeries O years thousands of surgeries 2012 596 a What is the meaning of N t N t is the rate at which the number of minimally invasive cosmetic surgery procedures performed in the country is changing with respect to time 9 148 10 927 O N t is the maximum number of minimally invasive cosmetic surgery procedures performed in the country between 2000 and 2014 ON t is the minimum number of minimally invasive cosmetic surgery procedures performed in the country between 2000 and 2014 ON t is the average number of minimally invasive cosmetic surgery procedures performed in the country per year ON t is the total number of minimally invasive cosmetic surgery procedures performed in the country between 2000 and 2014 11 551 13 065 b Construct a table of estimated values for N t Use a one sided difference quotient with an adjacent point for the first and last values and the average of two difference quotients with adjacent points for all other values Round your answers to two decimal places 13 955 N t 1x x 1x x in thousands of many invasive CUST x euc surgery x various yea x

a b C Find z if a 2 log 4 log 5 log 10 log b log 30 log 52 log c log 8 log a log a d log x 2 log 4 log 3x e log x 1 log 3 log x f ln x 1 1 86 log mn m logs TL log m log m log m log n log m log n alog m log if and only if d e f 711 11

Example 1 Find an equation of the tangent line to the function y at the point P 1 4 Solution We will be able to find an equation of the tangent line as soon as we know its slope m The difficulty is that we know only one point P on whereas we need two points to compute the slope But observe that we can compute an approximation to m by choosing a nearby point Q x 4x on the parabola as in the figure below and computing the slope mpo of the secant line PQ A secant line from the Latin word secans meaning cutting is a line that cuts intersects a curve more than once We choose 1 so that Q P Then mpQ For instance for the point Q1 5 20 25 we have the following The tables below show the values of mo for several values of x close to 1 The closer Q is to P the closer x is to 1 and it appears from the tables the closer mpo is to x 2 1 5 1 1 MpQ 50 32 5 18 564 0 5 1 01 16 242 199 MpQ 4 7 5 13 756 15 762 This suggests that the slope of the tangent line f should be m

10 81 4 2 6 4 1 2 2 4 2 A 4 6 8 10 12 6 Clear All Draw Dot Line Segment a Mark the two points on the graph that are needed to calculate Average Rate of Change over the interval 2 3 b Connect the points with a line segment c The Average Rate of Change over 2 3 is

A spherical balloon is being inflated Find the rate of increase with respect to its radius in inches r of the surface area in square inches S 4T when A r 2 inches Rate of increase in square inches per second B 4 inches Rate of increase in square inches per second C r 5 inches Rate of increase in square inches per second

Find equations of the tangent line and of the normal line to the curve y 5 2x 2 at the point 1 49 Write the equations of the lines in the form y mx b Tangent line y Normal line y