Application of derivatives Questions and Answers

often depe only the size of the population but also on how crowded it is When individuals have to compete for resources etc they cannot invest in reproduction leading to a decline in the rate of growth of the population which makes constant growth not sustainable Such models are usually called density dependent growth models One of such models is the logistic growth where the growth rate of the population depends only on the density of the population N and is given by the following 1 2 equation Where EXTRA LIST G N rN TN N K r 0 is a constant called intrinsic growth rate K 0 is a constant called the carrying capacity It represents the popula tion density that a given environment can sustain

Problem 1 Let f x 4x 2 and g x x 2 a Find the solutions to the equation f x 3 b Find the domains of f and g Argue why they are or are not injecti c Write fog d Find the inverse of f x e Calculate the first and second derivative of f x

Theorem 1 If f is a continuous function on the closed interval a b and differ entiable on the open interval a b then there exists a point c in the interval a b such that f c f b f a b a Try to use it to solve the next problem Problem 2 A car traverses a tunel which is 1km long in 50 seconds If the speed limit inside the tunel is 60 km h can we guarantee that the car went over the speed limit

The displacement in feet of a certain particle moving in a straight line is given by 8 t3 6 where t is measured in seconds a Find the average velocity over the given time periods 1 1 3 2 1 2 3 1 1 5 4 1 1 1 b Find the instantaneous velocity at t 1 1 2 3 4 b ft s ft s ft s ft s ft s

The point P 1 1 2 lies on the curve y x 1 x a If Q is the point x x 1 x find the slope of the secant line PQ correct to four decimal places for the following values of 1 5 2 9 4 999 3 99 5 1 5 6 1 1 7 1 01 8 1 001 1 2 3 4 5 6 7 8 LLL b Guess the slope of the tangent line to the curve at P Slope c Using the slope from part b find the equation of the tangent line to the curve at P

3 2 f x 4x 3x

r 2 X x 3 3 2 X x dx 4x 4

Sketch the graph of a differentiable function y f x through the point 3 3 if f 3 0 and satisfies the conditions given below a f x 0 for x 3 and f x 0 for x 3 b f x 0 for x 3 and f x 0 for x 3 c f x 0 for x 3 d f x 0 for x 3 a Choose the correct graph below OA Ay OB N HINH FITT F LINT X ON C J TRITT BRIHT T TSUL HHET O 10

Answer the following questions about the function whose derivative is f x x 1 x 2 x 8 a What are the critical points of f b On what open intervals is f increasing or decreasing c At what points if any does f assume local maximum and minimum values

Unit 5 Identify the inflection points and local maxima and minima of the function graphed to the right Identify the open intervals on which the function is differentiable and is concave up and concave down Question 3 of 8 Find the inflection point s Select the correct choice below and if necessary fill in the answer box to complete your choice OA The point s is are Type an ordered pair using integers or fractions Simplify your answer Use a comma to separate answers as needed OB There are no inflection points H This quiz 28 point s possible This question 4 point s possible G 2

K Find the value or values of c that satisfy the equation f c in the conclusion of the mean value theorem for the given function and interval f b f a b a kx x 3 3 10 Simplify your answer Use a comma to separate answers as needed Submit quiz CITED

xp x so x us 7 2

e 1 5 f dx ex

11 The scalar projection of a onto b is 8 If the length of a is 11 then what is the angle to the nearest degree between a and b

The function mis given by m x 36 2 2 Which of the following expressions could also define m x Jhry B 62 6 62 18 18 362

The function h is given by h x 5 3 1 2 What is the value of h 1 A B 5 3 15 5 9 4283

Where did they do it Hint Find the position at each time to the right t 0 On a playground t 1 5 At a baseball game The position of a vehicle moving in a straight line is Then s t t 3t 6 At which time is the particle determine moving towards the origin Why did they do it t 2 At Tony Stark s house velocity t 3 In a library The velocity of a particle is v t t 11t 2 Is the particle speeding up or slowing down at t 3 Slowing down bc a 3 0 Hates his costume Slowing down bc v 3 0 They are a Superman fan Speeding up bc a 3 v 3 0 They think spiders are creepy Speeding up bc a 3 0 Spiderman made fun of them

lim x40 e5x 1 3x

megd Name Pd2 Date 12 19 3 Your student email account requires you to choose a password of 4 letters followed by 1 Hour

Problem 1 1 point On which interval is f x 1 e6r decreasing A 0 B C 3 1 D 0 0 E 1 2 00

Problem 9 1 point Suppose the line y 8 r 1 is tangent to the curve y f x at x 5 If Newton s method is used for the approximation of a solution to f x 0 and the initial approximation is 5 find the second approximation 22 A 1 B 8 C 1 8 D 1 E 0 Problem 10 1 point Find the equation of the tangent line at defined by the equation y x x 1 0 B y 2 2 A Y E y 1 1 21 01 1 1 2 4 720 1 2 4 2 C y 2 2 1 2 2 1 X 2 2 to the curve D y 2 1

Find two positive numbers whose product is 143 and whose sum is a minimum

Find f x and find the value s of x where the tangent line is horizontal f x x 14x 63 f x X 7 7 2 x 14x 63 Select the correct choice below and fill in any answer boxes in your choice O A The value s of x where the tangent line is horizontal is Use a comma to separate answers as needed OB The tangent line is never horizontal

For the function below find A f x B the partition numbers for f C the critical numbers of f 7 X 4 f x A X B Find the partition numbers for f Select the correct choice below and if necessary fill in the answer box to complete your choic OA The partition number s is are x OB There are no partition numbers C Find the critical numbers of f Select the correct choice below and if necessary fill in the answer box to complete your choice OA The critical number s is are x OB There are no critical numbers Use a comma to separate answers as needed Use a comma to separate answers as needed

Awater sprinkler sends water out in a circular pattern Determine how large an area is watered The radius of watering is 17 ft Use 3 14 for OCEED The area of the circle is Type a whole number or decimal rounded to the nearest hundredth as needed

Find the profit function if cost and revenue are given by C x 130 2 4x and R x 4x 0 01x The profit function is P x

If 1000 is invested at 7 5 compounded continuously graph the amount in the account as a function of time for a period of 9 years Choose the correct graph A OD AA 2000 1000 0 4 0 2 4 6 8 10 AA 2000 1000 0 02 4 6 8 10 Q Q Q Q 5 OB OE AA 2000 1000 0 02 4 6 8 10 AA 2000 CEED 1000 0 14 02468 10 Q Q Q Q O C OF AA 2000 1000 0 0 2 4 6 8 10 AA 2000 1000 0 0 2 4 6 8 10 Q

Find dy for the function y dy Simplif 650 x

Use geometric formulas to find the unsigned area between the graph of y f x and the x axis over the indicated interval f x x 10 2 11 The area between f x and the x axis is

Give the order of the differential equation below where y represents a function of the variable x x y 2y y 2 Choose the correct answer below 9 OA The given equation is a second order differential equation because it involves a second derivative but no higher derivat OB The given equation is a third order differential equation because it involves a third derivative but no higher derivative OC The given equation is a first order differential equation because it involves a first derivative but no higher derivative OD The given equation is a first order differential equation because it only involves a first derivative

Write the following expression in the form ax bx9 where a and b are real numbers and p and q are rational numbers 8 8 x 3 2 8 8 x Simplify your answer Use integers or fractions for any numbers in the expression

A model for a company s revenue from selling a software package is R p 2 5p 850p where p is the price in dollars of the software What price will maximize revenue Find the maximum revenue The price maximizes the revenue Simplify your answer The maximum revenue is Simplify your answer

5 Detailed Clothing s total revenue R x and total cost C x in dollars from the sale of x custom sweatshirts are given as follows 6 pts R x 4x 11x 110 C x 4 2x 5x 10 a Find the profit function P x b Find the number of sweatshirts that must be produced and sold in order to yield the maximum profit c Find the maximum profit

11 Find the area between the curves y 4x x and y x 6x 8 Draw the graph and shade the appropriate region 7 pts 10 10 8 6 4 2 98 6 4 2 24 6 2 4 6 8 10

D Suppose an experiment finds that the memorization rate is given by M t 0 009t2 0 2t where M t is the rate in words per minute How many words are memorized in the first 10 minutes 5 pts

y 3x 5 and y 2x 3 No Solution Help 3 Submission Data 25 22222 24 23 21 20 19 18 17 16 15 14 13 12 3 11 10 9 8 7 Find the area between them from x 0 to x 3 Clear All Delete Fill WebAssign Graphing Tool

Find where the function is concave up or down include a sign chart and find inflection Points F x x 20 x

A drug taken orally is absorbed into the bloodstream at the rate of te 0 5t milligrams per hour where it is the number of hours since the drug was taken Find the total amount of the drug absorbed in mg during the first 6 hours Round your answers to two decimal places mg absorbed amount absorption rate mg Hours a Solve without using a graphing calculator 6 b Verify your answer to part a using a graphing calculator mg t

A boat sailing due West toward a point y 150 feet directly North of you makes an angle of 48 as shown in the diagram y 48 Find the distance in ft between you and the boat Round your answer to the nearest integer A

Sketch the graph of the function showing the amplitude and period y 5 cos 2t y 5 0 34 T O st t 2T y s DO 5 y M M 2T O st

Find the area between the curve y x 2 and the line y 2x 3 shown below from x 0 to x 3 y 11 9 7 5 53 3 square units 2 y x 2 y 2x 3 3

Find the general solution of the differential equation or state that the differential equation is not separable Enter NOT SEPARABLE if the equation is not separable y y 8x

Find the derivative of the function f x f x 4x 6x In x 1

F x x 3x 2x 8x 4 X4 3 Find first second third derivative

Aboat is being pulled toward a dock as shown in the figure to the right If the rope is being pulled in at the rate of 4 feet per second how fast is the distance between the dock and the boat decreasing when it is 3 feet from the dock CO Boat Rope The distance between the dock and the boat is decreasing by feet per second when the boat is 3 feet from the dock Round to the nearest thousandth as needed 3 ft Dock

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s x 3e2ln x 3 22 1 4

q x x 4x 5 2 3

limx 0 32 22 x

2x 1 limx 3 1