Application of derivatives Questions and Answers

b The graph of y g x is shown Draw the graph of y g K 8 6 4 8 6 4 2 6 6 8 X X 2

Score 11 17 11 17 answered Question 5 V A phone company charges for service according to the formula C n 19 0 04n where n is the number of minutes talked and C n is the monthly charge in dollars The rate of change in this equation is The initial value in this equation is Question Help Video Message instructor Select an answer Select an answer O

100x 0 4x2 where x is the The total revenue for Fred s Estates LLC is given as the function R x number of rooms rented What number of rooms rented produces the maximum revenue

Question 1 In 2002 a school population was 1933 By 2006 the population had grown to 2189 1 How much did the population grow between the year 2002 and 2006 students 2 How long did it take the population to grow from 1933 students to 2189 students years 3 What is the average population growth per year P Select an answer 4 What was the population in the year 2000 students 5 Find an equation for the population P of the school t years after 2000 6 Using your equation predict the population of the school in 2012

e 2 t 3 e 8 The answer above is NOT correct Find the solution to satisfying dy dt e t 3 e8 2y View y 4 3

c 5 pts Find the equation for the tangent line to f x at the point 1 3 x 2 obr tangent V

Chariots Mecklenburg Schools Practice Small group Instruction 1 Lin and Noah are solving the equation 7 2 2 91 Lin starts by using the distributive property Noah starts by dividing each side by 7 a Show what Lin s and Noah s full solution methods might look like b What is the same and what is different about their methods 2 A family of six is going to the fair They have a coupon for 1 50 off each ticket If they pay 46 50 for all their tickets how much does a ticket cost without the coupon Write an equation to find the ticket cost without a coupon

Problem 1 Solve each logarithmic or exponential equation a e3t 11e7 t 0 b log5 2 19 logs x 5 1 c 2x 2x 23

adc m the en 3 When making their fruit juice Meghan s Deli mixes 3 parts grapefruit juice with 2 parts apple juice If Meghan uses 12 cups of apple juice how much juice will she make

y 1 List all steps of differentiation the functions 4 2x 5 4 x 1 4 and y 2x 5

X y 20 99 Below answer parts a through f 17 15 119 117 Choose the correct graph below O A 150 00 14 129 10 139 O B 30 150 90 G d Use a graphing utility to find the line of best fit y 3 8613 x 179 292 Type integers or decimals rounded to four decimal places as needed e What is the correlation coefficient r r Type an integer or decimal rounded to three decimal places as needed C 30 150 OD

1 Determine the value of k for which f x 4x kx 6 has a horizontal tangent at x 2 3

1 Determine the derivative of each of the following functions Leave your answers in unsimplified form a y 3x5 7x4 b h t 5t 2 4t 3 1 c g x 2t 3 3t2 2t 1 2 3 3

8 The amount of pollution in a certain lake is P t 1 3 where t is measured in years and P is measured in parts per million ppm At what rate is the amount of pollution changing after 16 years

2 What are the coordinates of the point on the graph of f x x 1 x 2 at which the tangent is parallel to the line with equation 3x y 1 0 3

2 Use the first principles formula to find the derivative of y x 6

1 What is the slope of the normal to y x 2 x 2 at 2 0

0 10 For which values in the domain is this graph decreasing Give your answer in interval notation Assume the graph extends to infinity in both directions Try again 1 00 Edit my response

10 For which values in the domain is this graph decreasing Give your answer in interval notation Assume the graph extends to infinity in both directions 110 Submit

x 2y 4 7x 2y 8 Find the solution Give the answer as an orde pair

No Calculator Be sure to show some work or give an explanation lems 8 1 If the line tangent to the graph of the function fat the point 1 7 passes through the point 2 2 then f 1 is A 5 B 1 A 0 2 If f 3 8 and f 3 4 then f 3 02 is approximately A 7 88 B 7 92 C 7 98 D 8 02 B C 3 19 D 7 3 A particle moves along the x axis so that at time t 0 its position is given by x t 2t 21t 72t 53 At what time t is the particle at rest A t 1 only B t 3 only C t only 0 24 2 724 53 Name Shannon Dineen Date 10 03 23 B 5 5 If f is a continuous function defined by f x E Undefined 24 1 1 19 t 1 58 t 1 0 4 1 2 19 53 0 10242 19 53 0 421 4 What is the slope of the line tangent to the curve 3y 2x 6 2xy at the point 3 2 c 7 D E C 4 E 8 08 D t 3 and t 1 1 7 2 x bx x 5 5 sin Ex x 5 D 4 E 3 and t 4 then b E 5

The figure shows graphs of f f f and f Identify each curve a b c d bv f av f bv f YA X

The graph shows the influence of the temperature T on the maximum sustainable swimming speed S of Coho salmon SA cm s 20 0 10 20 T C What is the meaning of the derivative S T OS T is the maximum sustainable speed of Coho salmon for a given temperature OS T is the rate of change of the maximum sustainable speed of Coho salmon with respect to the speed of Coho salmon OS T is the rate of change of the maximum sustainable speed of Coho salmon with respect to the temperature OS T is the rate of change of the temperature with respect to the temperature OS T is the rate of change of the temperature with respect to the maximum sustainable speed of Coho salmon

X y 20 99 30 17 119 150 15 117 b Find the equation of the line containing the first and the last data points y 4x 179 Type an equation Type your answer in slope intercept form Use integers or fractions for any numbers in the equation c Graph the line found in part b on the scatter plot Choose the correct graph below OA 0 14 129 10 139 OB 30 150 001 0 ON O C 30 150 00 O D 30 156

The number N of locations of a popular coffeehouse chain is given in the table The numbers of locations as of October 1 are given Year 2004 2005 2006 N 2007 2008 8571 10245 12625 15015 16682 a Find the average rate of growth between each pair of years locations year 2004 to 2006 2006 to 2007 2005 to 2007 locations year locations year b Estimate the instantaneous rate of growth in 2006 by taking the average of the last two rates of change in part a locations year c Estimate the instantaneous rate of growth in 2006 by measuring the slope of the secant line through 2005 10245 and 2007 15015 locations year d Estimate the instantaneous rate of growth in 2007 by measuring the slope of the secant line through 2006 12625 and 2008 16682 locations year Compare the growth rates you obtained in part c and d What can you conclude O The rate of growth is decreasing O There is not enough information O The rate of growth is constant

For f x a fog 6 X 5 and g x b gof b g of x 5 find the following composite functions and state the domain of each c fof X Type your answer innervai notation Use integers of actions for any numbers in the expression your answer c f of x 5X 25 6 Simplify your answer Use integers or fractions for any numbers in the expression d gog The domain of g ofis 5 U 5 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression your answer 6 x 5 31 5x Simplify your answer Use integers or fractions for any numbers in the expression 31 The domain of fofis co 5 U 5 5 31 5 Type your answer in interval notation Use integers or fractions for any numbers in the expressic your answer d go g x Simplify your answer Use integers or fractions for any numbers in the expression

Find an equation of the tangent line to the curve at the given point y 49 7 To find the equation of a line we need the slope of the line and a point on the line Since we are requested to find the equation of the tangent line at the point 49 7 we know that 49 7 is a point on the line So we just need to find its slope The slope of a tangent line to f x at x a can be found using the formula f x f a Ha mtan lim G In this situation the function is f x Therefore we have y mtan lim x 7 lim 249 249 Using the factorization 49 7 x 7 we have x 7 2 49 7 x 7 So we found the slope and we know that 49 7 is a point on the line Substituting these values into the point slope formula y yo m o and solving for y we can conclude that the equation of the tangent line to f x at x 49 is zia f x f a HIP man lim and a

For f x a fog 6 X 5 5 and g x find the following composite functions and state the domain of each X b gof c fof 5 1 x Simplify your answer Use integers or fractions for any numbers in the expression d gog The domain off og is 0 U 0 1 U 1 0 Type your answer in interval notation Use integers or fractions for any numbers in the expression S your answer b gof x 5X 25 6 Simplify your answer Use integers or fractions for any numbers in the expression The domain of g ofis 5 U 5 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression S your answer c f of x 6 x 5 31 5x Simplify your answer Use integers or fractions for any numbers in the expression The domain off of is Type your answer in interval notation Use integers or fractions for any numbers in the expression Si your answer

Strontium 90 is a radioactive material that decays according to the function A t Age 0 024 amount present and A is the amount present at time t in years Assume that a scientist has of strontium 90 a What is the decay rate of strontium 90 b How much strontium 90 is left after 20 years c When will only 200 grams of strontium 90 be left d What is the half life of strontium 90 a The decay rate of strontium 90 is Type an integer or a decimal Include the negative sign for the decay rate

7 grams Do not round until the final answer Then round to the nearest thousandth as needed

K For f x a fog 6 X 5 5 and g x find the following composite functions and state the domain c x b gof c fof d gog a fog x 6x 5 1 x Simplify your answer Use integers or fractions for any numbers in the expression The domain of fog is 0 U 0 1 U 1 00 Type your answer in interval notation Use integers or fractions for any numbers in the exp your answer b g of x 5X 25 6 Simplify your answer Use integers or fractions for any numbers in the expression The domain of g of is Type your answer in interval notation Use integers or fractions for any numbers in the exp your answer

Write a balanced equation for the double replacement precipitation reaction described using the smallest possible integer coefficients A precipitate forms when aqueous solutions of copper II iodide and potassium hydroxide are combined Be sure to include states such as s or aq

Solve the equation by completing the square 1 3 4 64 0 How is the equation solved by completing the square Select the correct choice below and if necessary fill in the answer box es within your choice O A Use the formula x b b 4ac 2a with a b and c OB Rewrite the equation so the constant term is on the right side Add to both sides and factor the left side as a perfect square Then take the square root and is OC Write the equation in the form x p by moving the constant term and x term to the right side of the equation Then take the square root to solve for x OD Set the equation equal to 0 and factor the nonzero side of the equation Set each factor equal to 0 and solve

Ay y 20 2 3 3 5 4 6 20 c Graph the line found in part b on the scatter plot Choose the correct graph below O A 5 9 17 6 11 7 8 13 15 O B 20 0 20 d Use a graphing utility to find the line of best fit y 2x 1 Type integers or decimals rounded to four decimal places as needed 20 20

x 6x 55 How is the equation solved by completing the square Select the correct choice below and if necessary fill in the answer box es within your choice O A Use the formula x 2 b b 4ac 2a with a b and c OB Add to both sides and factor the left side as a perfect square Then take the square root and isolate x 2 OC Write the equation in the form x p by subtracting from both sides and then take the square root OD Set the equation equal to 0 and factor the nonzero side of the equation Set each factor equal to 0 and solve

Solve the equation by completing the square 1 12 3x X O A 0 How is the equation solved by completing the square Select the correct choice below and if necessary fill in the answer box es within your choice Use the formula x b b 4ac 2a with a b and c OB Rewrite the equation so the constant term is on the right side and the coefficient of x is equal to 1 Add take the square root and isolate x to both sides and factor the left side as a perfect square O C Write the equation in the form x p by moving the constant term and x term to the right side of the equation and then dividing both sides by solve for x OD Set the equation equal to 0 and factor the nonzero side of the equation Set each factor equal to 0 and solve Take the square rota

Evaluate OF f x dx where f x 62 T I 7 sin x 0 I T

Evaluate the indefinite integral 4 2 6 7 2 1 d dx C

T 4 7 cos u BUJ 16 du

Compute the indefinite integral 1 sin cos z

Evaluate the indefinite integral 224 47 dx C

O 6 5 b f x x 2 x 5 3 O f x x 2 x 5 O f x x 2 x 5 O f x x 2 x 5 2 T 7 6 5 4 3 2 1 1 2 3 4 Shift Mouse Wheel to zoom Shift Mouse Drag to pan 5 3 X x

Find the derivative of y O Select the correct answer below O O 3x 30x 25x 5 6x x 25x 5e x 6x 1 2 A 25x e x x 1 25x 5x

The slope field for the equation dP dt 0 0333333P 30 P for P20 is shown below 21 1 1 717 11 11 11111 38 On a print out of this slope field sketch the solutions that pass through 0 0 3 12 12 3 14 5 3 6 36 and 6 30 For which positive values of P are the solutions increasing Increasing for Give your answer as an interval or list of intervals e g if P is increasing between 1 and 5 and between 7 and infinity enter 1 5 7 Inf For what positive values of P are the solutions decreasing Decreasing for Again give your answer as an interval or list of intervals e g if P is decreasing between 1 and 5 and between 7 and infinity enter 1 5 7 Inf What is the equation of the solution to this differential equation that passes through 0 0 P 0 If the solution passes through a value of P 0 what is the limiting value of Past gets large

The surface given in spherical coordinates as cos p p is a cylinder

he local and absolute extreme values of f r s c d P q a b Q Give the local extreme values of f Choose correct answer below OA There is no local extrema O B Local minima are q and s local max are b and d O C Local minima are b and d local maxima are q and s Give the absolute extreme values of f Choos correct answer below OA Absolute minimum is q OB Absolute minimum value is b OC There is no absolute extreme value

For the following exercise solve the rational exponen x x 6x 6x 0 23 x Question Help VIDEO Written Example Post to forum Submit

Find f 4 for f x given below 4x 1 if x 4 f x 4x if 4 x 6 4 3x if x 6 OA 5 OB 8 O C 16

Current Attempt in Progress Which is the definition of the derivative of f x f x lim h 0 f x lim h 0 x h h 7 x h 7hx f x lim h 0 h 7 x h 7x h f x lim h 0 Of x 21x 7 xh 7hx h 7x

Examples 1 Find the slope of the tangent to the ellipse x 4y 25 at the point 3 2 2 The equation 5x 6xy 5y 16 represents and ellipse dy dx a Find at 1 1 b Find two points on the ellipse at which the tangent is horizontal