Application of derivatives Questions and Answers

2 A function is continuous on the closed interval 3 3 such that f 3 4 and f 3 1 The functions f x and f x have the properties given in the table below x f x f x 3 1 x 1 1 1 x 1 1 3 Positive Fails to exist Negative Zero Negative Positive Fails to exist c On the axes provided sketch a graph of f x Positive Zero Negative a What are the x coordinates of absolute maximum and minimum points of on the interval 3 3 Justify your answer b What are the x coordinates of all points of inflection on the interval 3 3 Justify your answer ty

Find all relative extrema of the function Use the Second Derivative Test when applicable If an answer does not exist enter DNE f x x4 8x 1 relative maximum relative minimum x y x y

3 Determine the intersection if any between the following three planes Give geometrical interpretation of the situation Explain how many solutions the situation has one unique solution infinite solutions or no solution The plane that contains the points R 2 1 3 S 1 2 1 and T 1 5 4 The plane with the normal 1 1 1 and that contains the point 2 1 2 The plane 2 4 4 s 1 3 0 0 1 1

If the graph of a polynomial function has three x intercepts then it must have at least two points at which its tangent line is horizontal True False

2 A small aircraft flies N 37 W with a constant ground speed of 300 km h The wind velocity is 80 km h N 75 E Determine the aircraft s air speed to one decima place and its heading using true bearing to the nearest degree

Find a b c and d such that the cubic function f x ax3 bx cx d satisfies the given conditions Relative maximum 2 9 Relative minimum 4 3 3 6 Inflection point a b C d

The graph of the derivative of f is shown below Assume that the domain of f is all real numbers and that it is a continuous function a Where is f x increasing 1 b Where is f x decreasing c Identify any local maxima of f x d Identify any local minima of f x e Where is f x concave up f Where is f x concave down g At what value s of x do points of inflection occur 4 3 1 1 2 3 4 5

f x DETAILS Determine the open intervals on which the graph is concave upward or concave downward Enter your answers using interval notation If an answer does not exist enter C 6 x 4 concave upward concave downward Need Help LARCALC11 3 4 013 Read It MY NOTES ASK YOUR

Which of the following cases is f x guaranteed to have both a maximum and minimum Select all that apply Select 1 correct answer s f x is continuous on a b and differentiable on a b f x is differentiable on a b f x is continuous on a b f x is continuous on a b

Consider the function f x x2 on the interval 3 3 Select all that apply Select 3 correct answer s The absolute maximum is 9 The function has no absolute maximum The absolute minimum is 0 The function has no absolute minimum The extreme value theorem applies on the interval 3 3 The extreme value theorem does not apply on the interval 3 3

Find the point of inflection of the graph of the function If an answer does not exist enter DNE f x 6 sec x 0 4x x y Describe the concavity Enter your answers using interval notation If an answer does not exist enter DNE concave upward concave downward

Consider the function on the interval 0 2 f x 3 sin x 3 sin x a Find the open interval s on which the function is increasing or decreasing Enter your answers using interval notation increasing decreasing b Apply the First Derivative Test to identify all relative extrema relative maxima x y smaller x value relative minima x y x y x y larger x value smaller x value larger x value

Determine whether the planes are parallel perpendicular or neither 18x 3y 9z 2 2y 12x 6z O parallel O perpendicular Oneither If neither find the angle between them Use degrees and round to one decimal place If the planes are parallel or perpendicular enter PARALLEL or PERPENDICULAR respectively

Find an equation of the plane The plane through the point 7 6 5 and parallel to the plane 7x y z

Let f x x 3x 3x2 24x 24 a Use the definition of a derivative or the derivative rules to find f x b Use the definition of a derivative or the derivative rules to find f x For the next parts of the problem used closed interval notation to enter your answers c On what interval is f increasing or more specifically non decreasing interval of increasing d On what interval is f decreasing or more specifically non increasing interval of decreasing e On what interval is f concave downward include the endpoints in the interval interval of downward concavity f On what interval is f concave upward include the endpoints in the interval interval of upward concavity

The graph of g is given below A Notice this is the graph of the derivative of g not the graph of g itself Answer the following questions about the graph of g based on the graph of g above A State the x values over which g is positive B State the x values over which g is negative C State the x values over which g is zero

Consider the function on the interval 0 2x f x sin x cos x 4 a Find the open interval s on which the function is increasing or decreasing Enter your answers using interval notation increasing decreasing b Apply the First Derivative Test to identify all relative extrema relative maxima smaller x value larger x value relative minima x y x y x y smaller x value larger x value

T Find T N and x for the plane curve r t 7ti 7 In cos t j 2 T t cost i sint j Type exact answers using radicals as needed N t sin t i cost j Type exact answers using radicals as needed cos t k t 7 Type an exact answer using radicals as needed

ind the arc length parameter along the given curve from the point where t 0 by evaluating the integral s t SIV 2 0 The arc length parameter along the curve starting at t 0 is s t Type an exact answer using radicals as needed The length of the indicated portion of the curve is L Type an exact answer using radicals and x as needed T sts 2 dt Then find the length of the indicated portion of the curve r t 8costi 8sint j 3t k where 0 sts

the graph of the function f x x2 x 12 a Find the equation of the secant line joining the points 2 6 and 4 0 b Use the Mean Value Theorem to determine a point c in the interval 2 4 such that the tangent line at c is parallel to the secant i c Find the equation of the tangent line through c d Use a graphing utility to graph f the secant line and the tangent line 10 06 2 10 8 6 4 2 ad Help Read to d 2 2 4 19 12 2 7 2 6 8 10 10 10 86 10 149 T 2 of

At 9 13 A M a sports car is traveling 35 miles per hour Two minutes later the car is traveling 81 miles per hour The mean value theorem guarantees that at some time during this two minute interval the car s acceleration is exactly A miles per hour squared Compute the value of A mi h

Use a graphing utility to graph the function on the closed interval a b f x 2x tan x Graph the function 02 02 0 1 0 5 0 5 0 5 05 02 02 02 0 1 0 5 No because f is not continuous on the closed interval a b No because fis not differentiable in the open interval a b No because f a f b 0 5 0 5 05 15 Determine whether Rolle s Theorem can be applied Select all that apply Yes 0 1 02 If Rolle s Theorem can be applied find the required values of c Enter your answers as a comma separated list If Rolle

Determine whether Rolle s Theorem can be applied to f on the closed interval a b Select all that apply f x x2 3 4 27 27 Yes Rolle s Theorem can be applied C No because fis not continuous on the closed interval a b No because fis not differentiable in the open interval a b No because f a f b If Rolle s Theorem can be applied find all values of c in the open interval a b such that f c 0 Enter your answers as a comma separated list If Roll Theorem cannot be applied enter NA

Consider the function f x 12x5 45x4 80x 5 For this function there are four important intervals o A A B B C and C oo where A B and Care the critical numbers Find A and B and C At each critical number A B and C does f x have a local min a local max or neither Type in your answer as LMIN LMAX or NEITHER At A At B At C

Use a half angle formula to find the exact value of sin 22 5 sin 22 5 3 X 6

Find the arc length parameter along the given curve from the point where t 0 by evaluating the integral s t SIV 0 dt Then find the length of the indicated portion of the curve r t 8cos ti 8sintj 3t k where 0 The arc length parameter along the curve starting at t 0 is s t Type an exact answer using radicals as needed The length of the indicated portion of the curve is L Type an exact answer using radicals and as needed CILE

The graph of g is given below Answer the following questions about the graph of g based on the information related to g and g A State the x values over which g is increasing B State the x values over which g is decreasing C State the x values where g has horizontal tangent lines D State the x values over which g is differentiable

An object moves in simple harmonic motion with period 7 seconds and amplitude 2 cm At time t 0 seconds its displacement d from rest is 0 cm and initially it moves in a negative direction Give the equation modeling the displacement d as a function of time t d J 3 16 Osin cos

x Find T N and x for the plane curve r t 7t i 7 In cost j t T i Oj Type exact answers using radicals as needed N i i Type exact answers using radicals as needed x t Type an exact answer using radicals as needed I

Graph the hyperbola 25 10 12 14 XOB X

Find the exact value of sin 0 5T by using a sum or difference formula 12 3 1

Minimum Wage Hour 1190765432 0 1984 Ontario Minimum Wage 1988 1992 1996 2000 2004 2008 2012 Year a Explain the general trend of the graph b How much has minimum wage increased from 1984 to 2008 What is the average increase p time period c If the minimum wage continues to increase at the same average rate predict the minimum wag 2020 d Predict when the minimum wage will r

8t y 25y e The general solution will be 5t y t c e t c e 5t

Find the derivative of the function F x F x x e X dt 869

et dt Find an equation of the tangent line to the curve y F x at the point with x coordinate 4 Let F x

The velocity function in meters per second is given for a particle moving along a line v t t 2t 8 2 t 6 a Find the displacement in meters 16 3 m b Find the distance traveled in meters by the particle during the given time interval 4 xm MY NOTES ASIC Enhanced Feedback Please try again Because the particle changes direction during the time interval the displacement will not be equal to the distance traveled The displacem velocity function The function for total distance traveled however treats any motion as positive so it is the integral of the velocity function s absolute vals Need Help Read It

If the units for x are feet and the units for a x are pounds per foot what are the units for da dx O feet per pound per pound denoted ft lb lb O product of pounds per foot and feet that is pounds O pounds per foot per foot denoted lb ft ft O product of pounds per foot and pounds denoted lb ft lb O product of feet per pound and pounds that is feet O pounds per foot per pounds denoted lb ft lb What unit does La x a x dx have O feet per pound per pound denoted ft lb lb O product of pounds per foot and feet that is pounds O pounds per foot per foot denoted lb ft ft O product of pounds per foot and pounds denoted lb ft lb O product of feet per pound and pounds that is feet O pounds per foot per pounds denoted lb ft lb

DETAILS 0 1 Points Find the general indefinite integral Use C for the constant of integration Remember to use absolute values where appropriate X PREVIOUS ANSWERS SCALCET9 5 4 015 x 2 4 log x x x C dx

0 1 Points DETAILS PREVIOUS ANSWERS SCALCET9 5 4 005 Find the general indefinite integral Use C for the constant of integration 5x4 6x 3 dx x 3x 3x C Need Help X Read it

Find the limit Hint Treat the expression as a fraction whose denominator is 1 and rationalize the numerator Use a graphing utility verify your result lim x x 4

Factor the GCF out of the following expression and write your answer in factored form 91x 0 39x5

Find the GCF of the following three monomials 39x5y 12x y 42x y

4 Find inverse of the matrix 1 3 3 A 1 4 3 C

Write the trigonometric expression in terms of sine and cosine and then simplify Ponoro cot 8 CSC 0 sin 0 COS 0

Soru 10 Puan A table of values for a function fis given below t f t A 7 0 1 B 14 2 2 Use 3 rectangles and right endpoints to estimate the value of the integral ff t dt 4 4 C 24 6 5 D 22 E 11

Soru 10 Puan Find the area of the surface generated by revolving the curve y 9 x 2 x 2 about the x axis A 12 B 48T C 6 4 D 24 E 16

Soru Evaluate the integral 0 x 4 dx 3TT C 3 16 A 10 Puan A B k D H8 E TU 16

Condsider the following system of equations 2 3 0 DO 670 J 7 2 LZ 6 0 Find the value of y using Cramer s Rule You don t need to find the values of x or Z

In Problems 11 13 sketch the graph of the function by finding its domain intercepts asymptotes intervals of increase or decrease local maximum and minimum values intervals of concavity and inflection points 11 y 3 x

64 Find the closets point on the parabola y x to the point 16 2