Differentiation Questions and Answers

Find the Maclaurin series of 6 7 0

Find the maximum and the minimum values of f x y z 6x 8y 4z on the sphere x y z 11 The maximum value is Simplify your answer

Find the maximum value of f x y 91 x y on the line x 9y 82 The maximum value of f x y 91 x y on the line x 9y 82 is Simplify your answer

Find three real numbers x y and z whose sum is 3 and the sum of whose squares is as small as possible The three numbers are Simplify your answers Use a comma to separate answers as needed

Find the radius of convergence for 6 x n n 8 n 1

Given the series 8 S k 0 6 does this series converge or diverge O converges O diverges 00 If the series converges find the sum of the series k 0 k 6 8 9 k If the series diverges leave this second box blank

Question A tree casts a shadow that is 34 feet long and the angle of inclination from the top of the shadow to the top of the tree is 39 How tall is the tree The tree the shadow and the dotted line in the image form a right triangle which means we can use a trigonometric function to help use figure out the height of the tree Select Select Select 39 Select 1 The length of the shadow is 34 feet so we plug that length into our formula and then solve for the height of the tree The height of the tree is foot

The reference angle is the angle measurement between the terminal side of the angle and the nearest half of the x axis Note that any angle s reference angle is always between 0 and 90 77 because the nearest half of 6 the x axis is only 30 away from the terminal side of the angle In the angle above the reference angle is 30 Question Select the image below that represents 1 a 300 angle in standard LOL 1

Example coefficient of g x 4x 2 5 2x From 4x 2 we need to look at the 4x but since 4x 2 is raised to the second power we need to raise 4x to the second power Similarly we look at the 2x from 5 2x and we raise it to the third power Multiply it all together 4x 2x 16x 8x 128 The degree is 5 and the leading coefficient is 128 Question Find the degree and leading coefficient of h x 3 5x x 7 2x 3 The degree is Select The leading coefficient is

A retail chain sells DVD players The price demand function of a particular model p x in dollars is given below for the weekly demand x A Describe how the graph of function p can be obtained from the graph of f x x B Sketch a graph of function p p x 117 3 x 9 x 289 now is px related to x O A The graph of p x is the graph of f x reflected in the x axis vertically shrunk by 1 3 B The graph of p x is the graph of f x reflected in the x axis vertically stretched by 3 and shifted up 117 units C The graph of p x is the graph of f x vertically stretched by 3 and shifted up 117 units OD The graph of p x is the graph of f x vertically shrunk by 3 and shifted up 117 units Choose the correct graph of p x O A Ap x 120 0 3 200 OB Ap x 200 of O 200 and shifted down 117 units Q O C Ap x 120 0 0 200 Q O D 200 Ap x 0 200

Solve the given equation for x Remember that e 0 for all values of x 4x e 64e 0 XE Simplify your answer Use a comma to separate answers as needed

The revenue in dollars from the sale of x car seats for infants is given by the following function R x 32x 0 010x 0 x 3200 A Find the average change in revenue if production is changed from 1 000 car seats to 1 050 car seats B Use the four step process to find R x C Find the revenue and the instantaneous rate of change of revenue at a production level of 1 000 car seats and interpret the results A Find the average change in revenue if production is changed from 1 000 car seats to 1 050 car seats Round to one decimal place as needed B R x C R 1000 R 1000 Interpret these results Choose the correct answer below OA This means that at a production level of 1 000 car seats the revenue is R 1000 dollars and is decreasing at a rate of R 1000 dollars per seat OB This means that at a production level of 1 000 car seats the revenue is R 1000 dollars and is increasing at a rate of R 1000 dollars per seat o in P 1 000 dollars and is increasing at a roto of B 1000 dollars por cont

A company estimates that it will sell N x units of a product after spending x thousand on advertising as given by N x 0 25x 22x 432x 76 000 where 18 x 48 When is the rate of change of sales increasing and when is it decreasing What is the point of diminishing returns and the maximum rate of change of sales Graph N and N on the same coordinate system The rate of change of sales is increasing on Type your answer in interval notation The rate of change of sales is decreasing on Type your answer in interval notation The point of diminishing returns is thousand dollars The maximum rate of change of sales is sales per thousand dollars Round to the nearest whole number as needed Graph N and N on the same coordinate system Choose the correct answer below O A 210000 N 1 OB Ay 210000 N OE M O C 210000 Ay 210000 N N in t O D M Q P

The number x of bicycle helmets a retail chain is willing to sell per week at a price of p is given by x 81 p 9 300 20 ps 100 Complete parts A and B Supply Demand 800 700 600 500 400 300 200 100 to 0 X X1000 491p 9 x 81p 9 300 N 0 20 40 60 80 100 120 Price Dollars A Find dx dp dx dp B Find the supply and the instantaneous rate of change of supply with respect to price when the price is 91 Interpret your results When the price is 91 the supply is When the price is 91 the instantaneous rate of change of supply with respect to price is At a price of 91 the supply is bicycle helmets per week and supply i at a rate of bicycle helmets per dollar

5 Use the implicit differentiation to find the derivative y given the implici equations cos 2xy x 2y dx

Calculate the sample standard deviation of the data shown Round to two decimal places X 15 17 222 14 19

d Find 6 Type In x for the natural logarithm function dx

Differentiate the following function You do not need to simplify the derivative h x 5e2 h x

A radar gun was used to record the speed of a runner during the first 5 seconds of a race see the table Use Simpson s Rule to estimate the distance the runner covered during those seconds Round your answer to three decimal places m S t s v m s 0 0 5 1 0 1 5 2 0 2 5 0 4 61 7 32 8 86 9 77 10 28 t s 3 0 3 5 4 0 4 5 5 0 v m s 10 57 10 63 10 78 10 89 10 89

A circular swimming pool has a diameter of 24 ft the sides are 6 ft high and the depth of the water is 5 ft How much work in ft lb is required to pump all of the water out over the side Use the fact that water w 62 5 lb ft3 Round your answer to the nearest integer 848230 ft lb

The antiderivative of f x is unique True False

nd and du Z Z V dv u In 3 v 1 u In 3 v 1 1 when u In 3 v 1 if z 10 tan 3 Simplify your answer 11 Simplify your answer x and x e In v

e given functions a express dw as a function of t both by using the Chain Rule and by expressing w in terms of t and differentiating directly with respect to t Then b evaluate dt JU w 10x 6y x cost y sin t t 2 dw dt at the given value of t

Assuming 8x 2y 6xy 0 defines y as a differentiable function of x use the theorem 0 0 une an integer or a simplified fraction dx X to find dx at the point 1 1

dz dz du Find and when u In 3 v 1 if z 10 tanx and x Inv dz dv du 11 Simplify your answer

Use the limit process to find the area of the region between the graph of the function and the x axis over the given interva y 4x 6 0 1 Sketch the region O y 3 2 2 3 y 3 y 3 2 4

dz se the functions below to a express and as functions of u and v both by using the Chain Rule and by dv dz dz expressing z directly in terms of u and v before differentiating Then b evaluate and du dv z 4 tan dz x 11 Next find y 4y 2 2 y X dz du z dy 4x 2 2 x y Express z directly in terms of u and v x u cos v y usin v u v Type an exact answer v 6 1 7 Type an exact answer K at the given point u v

Find the velocity acceleration and speed of a particle with the given position function r t t i 7tj 3 In t k v t a t v t B 2ti 5j t 2i 0j k 2 4 49 X 9 7016

6 What is the total resistance for the circuit below R 2927 R 352 R3 6925

9u u 3 du Describe the most appropriate substitution case and the values of u and du Select the correct choice below and fill in the answer boxes within your ch OA Substitute u for the quantity in the numerator Let v so that dv B Substitute u for the quantity in the denominator Let v C Substitute u for the quantity under the root Let v so that dv Use the substitution to evaluate the integral so that dv du du du

5 20 points Consider the second order differential equation y ay by f t where a and b are constants If initial conditions are given by y 0 yo y 0 to use Lap Transforms to find an expression for Y s Do not attempt to find y t T

Find equations of the tangent lines to the curve Y x L that are parallel to the line X 21 2 Enter your answers a X L a Comma Sepereted list of equations X 1 1 and Y Y 2 X 1 y G Y X and Y and X 1 2 X 1 2 X 1 2 y 7

An equation for the line tangent to the graph of the differentiable statements function fat x 3 15 Y 4x 6 Which of the following must be true 1 f 0 b 2 3 18 3 f 3 4 None 61 and 2 only O2 and 3 only 1 2 and 3

Find f x and f x for f x x e A f x 4x ex and f x 12x AX f x 4x ex x cx and f x 12 x e 4x e 4 e f x 4x e and f x 12x ex 4x e X 4 X 3x 2 f x 4x e x e and f x 12x c 8x e x e

Differentiale K 0 2x 1 x 2x Select the correct answer 3 6x 4x 2 14x 4x 12x 2x 3 2 D 6x x4 2 4x 2x 1 bx x xx 2x 1 4x 2 2 x x 2x 2x 1 4x 2

Let be a differentlable function such that f 3 2 ard F 8 5 If g 13 the function defined by g x f x 2 x What is the velve of g 8 359 24 2 61 24 60

Integrate the given series expansion of f term by term from zero to x to obtain the corresponding series expansion for the indefinite integral of f If f x 1 8nx8n 1 8x7 1 x8 f t dt 20 n 1 8 n 0

Find the future value for the ordinary annuity with the given payment and interest rate PET 750 1 30 compounded semiannually for 4 years The future value of the ordinary annuity is Do not round until the final answer Then round to the nearest cent as needed

Problem 12 Test the following three series for convergence n 22n n n 3 3 n 76 2 sin n 3n

2 Use the function g x 0 5 5 x x 1 2 for the following 1 1 2 1 marks a Determine the end behaviours of the function behaviour end c What is the value of those differences b Which differences would be the same d State where a bend or bounce occurs

2 25 pt Find the value of zyy at the point 1 1 0 where yz x y e

y 2 tan 2x 3 Plot the two most central vertical asymptotes as a vertical lines Then plot the 3 defining points between them using the dot tool k Q b 2 3 2 Period NT TU 2 in 5 4 3 2 1 77375 1 2 3 4 5 y TU 2 M 31 2

The graph of the rational function f x is shown below Use the graph to complete each limit statem Provide your answer below Asr x 0 0

Karl Yates needs 24 000 to pay for the remodeling work on his house A contractor agrees to do the work in 10 months What is the present value of 24 000 at 6 9 simple interest in order to accumulate the 24 000 in 10 months The present value is Round to the nearest cent as needed ECCO

For the function f x y sin 7x 8y find of and 513

a What point is on the graph of y g x 6 3 b What point is on the graph of y 2g x 4 2 c What point is on the graph of y g 2x 4 a What point is on the graph of y g x 6 3 Type an ordered pair b What point is on the graph of y 2g x 4 2 Type an ordered pair c What point is on the graph of y g 2x 4 Type an ordered pair

Find the value of az OzZ at the point 1 1 1 if the equation 4xy z x 4yz 1 defines z as a function of the two independent variables x and y and the partial derivative exists 07 1 1 1 Type an integer or a simplified fraction

Find ax ax and P5 dy 2 f x y 4x 5y Type an exact answer using radicals as needed

Find the slope of the line graphed below 0

Perform the following operation Express your answer in scientific notation What is the result of 3 0 x 104 x 9 0 x 104 A 2 7 x 107 B 3 0 C 2 7 x 10 D 27 x 108