Differentiation Questions and Answers

Suppose that f x g x and h z are differentiable functions Values for the functions and derivatives are shown in the table below I 1 0 1 1 1 g 1 g 1 h 1 1 1 3 4 1 0 1 3 3 2 2 2 d g x dr h r Compute the values given below a f x g x h x at x 1 b d f x g x at x 1 2 at x 0 d h x at x 1 dr 1 0 0 1 1 0

a Find dy dx y 9xy implicitly defines y as a function of x i e find y

Chapter 2 Test Page 3 of 10 1 March 2023 TRUE or FALSE For each statement below determine whether it is true or false by circling the ap propriate answer No work needs to be shown For all problems below z and y denote real numbers a d da 77 2 b da sin 3 sin 2 cos x 2x c The derivative of a polynomial is a polynomial d Every function has an inverse TRUE e If f x is continuous at x 2 then f x is differentiable at x 2 f If f x is differentiable at x 2 then f x is continuous at x 2 g Linear functions have a constant rate of change FALSE TRUE FALSE TRUE FALSE TRUE FALSE TRUE FALSE TRUE FALSE TRUE FAL

8 Find y by implicit differentiation simplify your answer completely 3 dy 4 45 T 45 1 x dy C 0 dy 4x df 4x f 3 45 9 15 12 x 4x 12 3 8 x 23 2 3 45 48 x y 453 2 3X 6 x y 2 x 3 6 5 9 1 9 1

14 If f is continuous on 0 and fi f x dx is convergent then f f x dx is convergent

Solve the equation x 3 x 3 x 4 x 4 F x 0 Enter your answers as a comma separated list x

If f x 6 sin x 12 cos x then f x

A graphing calculator is recommended For the limit lim x 5x 5 17 illustrate the definition by finding the largest possible values of 6 that correspond to 0 2 and r 0 1 Round your answers to four decimal p 0 2 8 M 8 m r 0 1

Flag question Question 5 Not yet answered Marked out of b 9 pour O c 9 foot d 225 fo Let u 4 5 ject one Select a 23

19 Find the second derivative of the function f x 2x 6

ake the derivative of each function 1 f x 1 x x

6 Let y x zz 3 6 and z 3 1 Find y 3

4x 2x 4 y 3x 5 5 Let f x g x h x g 10 4 h 10 560 g 10 0 and h 10 35 Find f 10

2 g x x x 1 2 3 f x

Take the derivative of each function 1 x 1 x x 2 g x x x 1 2 3 x 3 1 1 55 4x 2x 4 y 3x 5 5 Let f x g x h x g 10 4 h 10 560 g 10 0 and h 10 35 Find f 10 6 Let y x 2x z 3 6 and z 3 15 Find y 3

Use a calculator to approximate the length of the following curve Simplify the arc length integral as much as possible before finding an approximation r t 4 cost 12 sin t for 0 st 2 A L 32J cost dt 0 2x OB L 4 1 8 cos t dt 0 O C L 2x Sri 0 16 sin t 144 cos t dt 2x D L 32 sint cost dt 32 f s 0

If f x x 1 and g z z 1 a f g x b g f x c Thus nand

a Find the value of J and the material and spatial descriptions of the velocity for the motion given by X atX F1 x2 1 3t X 23 X3 where the constants t or and 3 satisfy t a 320 b Calculate V and y Du

Graphs Draw the graph of the function and state the domain and range f x 5x 3 Choose the correct graph

If f x 5 x ln x find f x Find f 2

If f x 7x 2 5x 5 find f x using logarithmic differentiation

Let Find the following derivatives dy dx d y dx 15x 4 4 4y 1 3x5 8y 4y 0 help formulas

rain gutter is made from sheets of aluminum that are 15 inches wide As shown in the figure the edges are turned up to form right angles Determine Flat sheet e depth of the gutter that will allow a cross sectional area of 11 square inches Show that there are two different solutions to the problem Round to 15 inches e nearest tenth of an inch wide he depth of the gutter is inches SITS 15 2x

23 Calc I nfidently in the WORK ON E 10 pts Like 3

JOU ed Salad Attempt 1 Question 3 0 5 points Listen Use Newton s method to determine the posit cos x x correct to four decimal plac

Determine whether the graph represents a function If and range O A Type an inequality or a compound inequality

wer the following angular speed questions Enter your answers using exact values a A wheel of radius 25 ft is rotating 13 RPM counterclockwise Considering a point on the rim of the rotating wheel what is the angular speed in rad sec and the linear speed v in ft sec 1 36 rad sec V b A wheel of radius 9 in is rotating 45 sec What is the linear speed v the angular speed in RPM and the angular speed in rad sec V 7 06 in sec V X ft sec 905 X c You are standing on the equator of the earth radius 3960 miles What is your linear and angular speed mph rpm rad sec d An auto tire has radius 12 inches If you are driving 65 mph what is the angular speed in rad sec and the angular speed in RPM rad sec rad hr X rpm

Solve the system of equations by the addition state whether the system is inconsistent or de 3x 4y 7 8y 6x 14 What is the solution to the system of equatio

Let f x 13 a 2x with f 0 16 and f 8 61 Find a and k and then find f 2 The value of a is The value of k is Type an integer or a simplified fraction Type an integer or a simplified fraction Find f 2 f 2 Type an integer or a simplified fraction

Attempt i on 5 0 5 points sten sum of two positive numbers is 50 Wha of their squares

eir Question Part Determine whether the graph is represent a function give its don

Round to three significant figures 17 54458 rounds to

1 Determine whether the graph represents a func range

Solve the equation in y 14x 74 0

a x C X x 4 O 6 b x x 27 and 3 ABC D 7x 6 4x 3 and x 545 E and X d x and x 5 3 e x x and and x St 6 x

suppose f x x 5 x 6ex 3 with remainder 16 Then f 5

polynomial of degree 4 has a root of multiplicity 2 at x 3 a root of multiplicity I at x 0 at x 2 5 20 it goes through the point find a formula for P x P x

4 3 The polynomial f x Sx 6x 12x wx S is divided by x 2 For what O value of w Will the remainder be 3 W

a thousands of pairs of sunglasses and produced P q is Potal profit Sold 5 Sunglass Sunglass Hut compart P 9 8 019 49 How many should be sold to maximize profits thousands HUGH HAM be pairs need to Sold What are the actual maximum profits that can be expected S

Interval of a function is called decreasing when y goes down decreases as x increases moving to the right Similarly an interval of a function is called increasing when y goes up increases as x increases moving to the right Not we always express these intervals in terms of x vlaues Based on above definition of increasing and decreasing intervals for a function Consider f x x identify the interval given below as increasing or decreasing interval infinity 0 is this an increasing or decreasing interval 0 infinity is this an increasing or decreasing interval

Macmillan Learning What is the chemical formula for the chemical formula Cr ClO3 3 What is the chemical formula for th

Determine the slope of the line th 3 2 and 7 9

Graph the equation and state the slope of the line

Solve the equation by factoring 15x 31x 10 0 Select the correct choice below a

sing the Solve the equation by factoring 2 x 10x 21

Determine where the second derivative of the function is positive and where it is negative Z O positive 5 x 3 and x 0 negative x 5 and 3 x 0 O positive x 1 and x 0 7 negative 1 x 0 7 O positive x 5 and 3 x 0 negative 5 x 3 and x 0 O positive 5 x 3 and x 1 negative x 5 and 3 x 1

Determine the critical numbers of the given function and classify each critical point as a relative maximum a relative minimum or neither f t 10f6 485 604 7 Ox 0 2 0 7 relative minimum 2 57 neither Ox 0 2 0 7 relative maximum 2 57 neither Ox 0 0 7 relative maximum Ox 0 0 7 relative minimum

Find dy dx by implicit differentiation x y3 5 O x 2 O y x Oy x x

Find the relative extrema if any for the function f x x3 3x2 24x 6 local max at x 4 and local min at x 2 local max at x 4 O local max at x 2 and local min at 4 local min at x 2

Find y for y y x defined implicitly by 3xy x2 4 0 O y 3 x Oy 3x 2y y 4 O y 3y 2x 3x O y 2x 3y 3x