Application of derivatives Questions and Answers

2 In this problem you will investigate the family of functions h x x cos ax where a is a positive constant such that 0 a 4 A Graph the curves y x x cos x and y x x cos 4x in the space provided below 4 B For what values of a will h x have a relative maximum at x 1 C For what value s of a will h x have an inflection point at x 1 I

Note t 0 corresponds to midnight Time is measure in hours and the rate of water usage is measured in millions of gallons per hour Rate millions of gallons hour Time 0 Rate 0 5 1 70 1 55 1 40 1 25 1 10 0 95 0 80 0 65 0 50 0 00 2 0 6 4 0 9 4 00 6 8 1 3 1 7 10 12 14 16 18 20 22 24 1 5 1 3 1 4 1 5 1 4 1 1 0 7 0 5 Water Usage Rate for Seattle WA Over 24 Hours 8 00 12 00 Time hours 16 00 20 00 24 00 24 A Using three trapezoids of equal width estimate frate t dt Explain the physical meaning of fratel t dt this definite integral B Does the estimate in part A overestimate or underestimate the value of frate t dt on the interval 0 t 8 Explain your reasoning a C Using your answer from part A estimate the average water usage over the 24 hour period

5 The figure at right is the graph of f x the second derivative of a function f x The domain of the function f x is all real numbers and the graph above shows f x for 3 6 x 3 6 30 20 10 A Find all values of x in the interval 3 6 3 6 where f x has a horizontal tangent B Find all values of x in the interval 3 6 3 6 where f x is concave upward Explain your answer

How many critical points does the function f x x 1 6 x 5 have O A 6 OB 13 O C 1 OD 7 OE 3

What is the average value of y for the part of the curve y 4x x that is in the first quadrant O A B 3 2 O c D 2 3 E 3 2 2

The slope of the line normal to the graph of y 2 ln sec x at x is OA 2 OB 1 2 OC 2 D nonexistent E OE 1 2

In an experiment to study the growth of bacteria a medical student measured 5000 bacteria at time 0 and 8000 at time 10 minutes Assuming that the number of bacteria grows exponentially how many bacteria will be present after 30 minutes A 14000 bacteria B 29333 bacteria C 24332 bacteria D 20480 bacteria E 17830 bacteria

The velocity of a particle on the x axis is given by the differential equation 3 da t and the particle is at x 5 when t 1 The position of the dt particle as a function of time is x t O A x t gt 2 5 B x t t 5 c x t 1 12 D x t t 5 E x t et 5

The area of the region enclosed by the curves y 2x and y x 4x is A 18 B 12 143 O C 12 D 14 E 36

If limx 3 f x 7 which of the following must be true I f is continuous at x 3 II f is differentiable at x 3 III f 3 7 OA None O B II only O C III only OD I and III only OE I II and III

Which of the following is a right hand endpoint Riemann sum approximation to f f x dx with six equal subdivisions A 1 5 2 f 2 5 f 3 3 5 4 O B O c f 1 f 1 5 f 2 f 2 5 f 3 f 3 5 f 1 f 1 5 f 2 f 2 5 f 3 f 3 5 O D f 1 f 1 5 f 2 f 2 5 f 3 f 3 5 E 1 5 f 2 f 2 5 3 f 3 5 f 4

Which of the following curves exhibits modified exponential decay has a y of 110 when tis 0 and approaches the value y 30 as t goes to infinity OA y 110 80e 5t B y 30 80e 4t OC y 110 80e t OD y 30 80e 2t OE y 110e t 80

The rate of change of P with respect to tis directly proportional to 10 and inversely proportional to the cube root of P The differential equation that models this situation is O A d OB c de dt k 10 t 1 T kv P 10 t 10 t P dP O D dt k 10 t VP O E k 10 t P

The following graph could be the slope field plot for which differential equation below X 2 O A y Y Click here for long description You should try to do this without your calculator by examining the slope field in each of the four quadrants as well as on or near the axes O B y 1 yx O O D y O E y C y 1 2 1111111 Go I 1 1 1 1 1 1 1 1 1 11 11

Solve the following differential equation with initial conditions y e 2t 10e t y 0 1 y 0 0 The solution is 5 4t O A y e e 3t O B y e 2t 54 2 O C y 4e 2t 16e t 6t 19 OD y e 2t e4t 1 O E y 2t e t 2t 4t 8

he acceleration of a particle moving along the x axis at time tis given by a t 6t 2 If the velocity is 25 when t 3 and the position is 10 when t 1 the position x t A t t 4t 6 B 36t 4t2 77t 55 O c 9t 1 D t3 t 9t 20 OE 3t 2t 4

The family of curves represented on this graph would appear to be consistent with which of the following differential equations A d B D da Y Te tan x y d x y x x 3 E d 3x

Let f be the function defined by f x X Which of the following statements is true OA fis discontinuous at x 0 B fis an odd function O c f 0 0 OD f x 0 for x 0 E fhas a relative maximum x for for x 0 x 0

The maximum value of f x e x is OA 1 OB 1 OC 2 OD In 2 OE O

For the following function make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point f x 11 cos x at x RIN x 2 Complete the table below Round the final answer to three decimal places as needed Round all intermediate values to four decimal places as needed Slope of secant line Interval T

Which of the following differential equations is are not separable 1 t yt 1 dt II d y dx 3xy III dy y yx da O A I only O B II only O C III only D II and III dy dx OE I and III

Factor completely Then solve p w 3w 6w 24w

9 Discontinuities at 0 and 3 but continuous from the right at 0 and from the left at 3

If x 2xy y 2 at the point 1 1 dy is da A 0 1 OB B 2 O C 12 2 OD nonexistent OE

The temperature over a given period is Estimate the average temperature from 0 t 8 using the left endpoints of four equal subintervals Time hours 0 1 2 3 4 5 6 7 8 Temp F 35 38 45 40 37 22 14 8 13 A 26 4 F B 26 F C 50 F OD 32 75 F

Let f be a polynomial function with degree greater than 2 If a b and f a f b 1 which of the following must be true for at least one value of x between a and b I f x 0 11 f x 0 III f x 0 OA None OB I only O C II only OD I and II only OE I II and III

Consider the position function s t 4 9t 29t 17 Complete the following table with the appropriate average velocities and then make a conjecture about the value of the instantaneous velocity at t 4 Complete the following table Type exact answers Type integers or decimals Time Interval Average Velocity 4 5 4 4 1 4 4 01 4 4 001 4 4 0001

The slope of the line normal to the curve ex x 2 10 at the point 0 3 is OA O B 6 O C 6 O D 6 OE Undefined

2 Which of the following equations is perpendicular to the graph of 4x 3y 10 A 3x 4y 10 B 4x 3y 10 C 3x 4y 12 1 D 3y 12 4

3 6 A linear function contains the points 2 5 and 15 20 Does the point 43 53 also lie on the line Explain 2

4 Line 1 passes through the points 2 5 and 6 k and is perpendicular to the line given by y 1 x 1 5 Find the value of k

1 Write the equation of the line that is perpendicular to y 5x and passes through 7 2

The position of an object moving vertically along a line is given by the function s t 4 9t2 26t 17 Find the average velocity of the object over the following intervals a 0 7 b 0 4 c 0 1 d 0 h where h 0 is a real number

O A 2 ln sin 0 2 4 C OB In sin 0 2 4 C O c 2 ln cos 0 2 4 C O D In cos 0 2 4 C OE None of these s

g x x 2x 1 What is the slope of the tangent line to the curve g x at the point 2 1 O A 1 OB 5 OC 8 OD 14 OE E

z 2 x y is an equation for the tangent plane to z f x y at the point 0 0 Which of the following is the linear approximation for f 0 02 0 01 O2 03 1 97 1 99 O2 01

Suppose z 1 2x 3y is an equation for the tangent plane to z f x y at 0 0 Then we also know that select all that apply Of 1 1 0 Ofz 0 0 2 Of 0 0 3 Of 0 0 1 The vector n 2 3 1 is orthogonal to the tangent plane

dx O A OB 1 12x O D O E 1 x C None of the above 12 12x 4 1 12x 4

da In 12x 4 dx 1 12x O A OB OC None of the above O D 12 12x 4 OE X 1 12x 4

Let y In t tan a OA 2 tan x sec x O C B 2 tanx sec x tan x secr tan a O D OE Then y tan x x sec x a tan x sec x x tan x sec tan r

The curve y x log3 x 5 has points of inflection at x OA In 3 5 B There are no points of inflection O c 10 O D D 10 In 3 OE 5

da A 2 In 2 In x 2 2 x In 2 In x 2 O B B O C In 2 2 X O D D 2 In 2 2 In x 2 X O E In 2 2 In x 2 X

Let f x 1 x if x 0 2x 1 if x 0 Evaluate f 3 and f 6 f 3 8 f 6 If you have had difficulty with these problems you should look at Sections 1 1 1 3 of y

O A 22 In 2 In x 2 B In 2 In x 2 In 2 2 I OB O C OD In 2 2 ln x 2 O E In 2 2 In x 2

a y c 2 1 0 0 4 1 9 2 15 b d 3x x y 11

3000 young trout are introduced into a large fish pond The number of trout still alive after tyears is modeled by the formula N t 3000 0 9 What is the rate of population decrease when the number of trout in the pond reaches 2000 A Approximately 520 trout per year B Approximately 211 trout per year C Approximately 2000 trout per year D Approximately 494 trout per year OE None of the above

Find intervals where this function is increasing decreasing or constant 3 f x x 3 10 5 1 1 2 3 4 5 6 7 5 10 A increasing from to 3 and decreasing from 3 to o B decreasing from to 3 and decreasing from 3 to o C increasing from t3 and decreasing from 3 to co

Find the domain of the function f x x 2 A x22 C no solutions B x 2 D x 2

da O A O O B O c x 3 x ln 2 1 3x O D None of the above 1 O E x In 2

dx O A OB B e3r 1 e z 1 2 2 1 e 1 x 1 2 O c 1 1 2 O D 3 D 2 1 2 E 3 1 x 1 2