Application of derivatives Questions and Answers

Convert the following to rectangular coordinates Angles are given in radians 5 pts each A B Give the exact value Give rounded to 1 decimal place 6 2

During a figure skating routine Jackie and Peter skate apart with an angle of 15 between them Jackie skates for 5 meters and Peter skates for 7 meters How far apart are the skaters

From the top of a 108 foot lighthouse the angle of depression of a boat at sea is 27 Find the horizontal distance from the boat to the base of the lighthouse to the nearest foot A MathBits com 27 C AN B

10 Copy the following sign chart of y x to your work and draw a curve y f x above it whose 1st derivative fits the sign chart y 0 at a b c and d 11 Copy the following sign chart of y x to your work and draw a curve y f x above it whose 2nd derivative fits the sign chart y 0 at a b and c C

2 Find the exact value Do not use a calculator Write undefined if it does not exist Show rough sketch of unit circle or bow tie as needed sin sin 1 2 sin sin sin 3

You roll a 6 sided die two times What is the probability of rolling an odd number and then rolling a prime numb Write your answer as a percentage

4 Assume x x t y y t and z z t and that sin xy x 1 y Expand the derivatives d dt as much as possible then solve for dx dt in terms of x y and dy dt

3 Assume y y x a function of x find dy dx and where the tangent is horizontal and where the tangent is vertical if x y x y if any

1 Find the derivatives a y tan x sec x b y tan sec x 1 2 Assume y y x a function of x find dy dx if exy In x y 1

Find the solution to the system of equations x 2y z 2 y 22 3 x y 3z 7 I y 2 Question Help Video Message instructor

5 Assume x x t y y t and z z t and that x y z 100 find dz dt if x 6 y 7 dx dt 2 and dy dt 5

Solve the following system Y 3y gy 4z 12z O Exactly one solution O Infinite solutions No solution 4 9 30 Question Help Message instructor

Fill in the blank with an appropriate word phrase or symbol s A closed figure in a plane determined by three or more straight line segments is called a A closed figure in a plane determined by three or more straight line segments is called a polygon regular congruent figure

9 A backpack selling company determines that if it sets the price of its daypacks is set at p dollars then it can sell q 400 000 8 p daypacks per year A Find the Revenue Function B What price should the daypacks be priced at to maximize revenue aximum Revenue

Determine if the requirements for Rolle s theorem are met by the function f x 52 or on the x interval 3 3 If so find the values of cin 3 3 guaranteed by the theorem f x is continuous on 3 3 but not differentiable on 3 3 The conditions of Rolle s theorem are not met f x is not continuous on 3 3 Therefore the first requirement is not met and Rolle s theorem does not apply 5 f x is continuous on 3 3 and differentiable on 3 3 When evaluated f 3 and f 3 5 Therefore f a f b and the conditions of Rolle s theorem are met The value guaranteed by Rolle s theorem is c 0 lo f x is continuous on 3 3 and differentiable on 3 3 When evaluated f 3 55909 5 and f 3 Therefore f a f b and the conditions of Rolle s Theorem are not met

Find the first derivative You don t need to Simplify the algebra But Circle your answer use Implicit Differentiation x y sy

Determine if the conditions of the mean value theorem are met by the function f x 2x 4x on 2 2 If so find the values of c in 2 2 guaranteed by the theorem O O f x is a polynomial and therefore is continuous on 2 2 and differentiable on the interval 2 2 The values guaranteed by the mean value theorem are c 2 3 The values guaranteed by the mean value theorem are c and c f x is a polynomial and therefore is continuous on 2 2 and differentiable on the interval 2 2 2 3 3 6 and c 3 6 f x is a polynomial and therefore is continuous on 2 2 and differentiable on the interval 2 2 The value guaranteed by the mean value theorem is c 6 3 f x is a polynomial and therefore is continuous on 2 2 and differentiable on the interval 2 2 O The value guaranteed by the mean value theorem is c 2 3 3

From the graph of f x determine the graph of f x 12 10 8 6 2 y 12 10 8 6 4 2 0 2 4 6 8 10 12 2 6 8 10 12 x

For f x A Domain B Asymptotes C y and x intercepts D Where the function is increasing or decreasing E Concave down or concave up F Relative extrema G Inflection points H Graph of the curve x4 x3 find the following Carefully verify all of the information is correct when making your selection A Domain is all reals B No asymptotes C y intercept 0 0 x intercepts are 0 0 0 0 0 4 0 D Decreasing on 3 increasing on 3 00 E Concave up on 3 concave down on 3 0 concave up on 0 00 F Relative min of 6 75 at x 3 G Inflection points at 2 4 and 0 0 H

Find all the critical numbers of f x 2x4 4x 3x 2 then determine the local minimum and maximum points by using a graph O O O Critical number x 1 There is no local maximum or local minimum Critical numbers x 1 and x 1 There is no local maximum or local minimum Critical numbers x 1 x 0 x 1 0 2 is a local and global minimum Critical numbers x 0 and x 1 0 2 is a local minimum

A farmer plans to fence a rectangular grazing area adjacent to an existing fence The grazing area must contain 180 000 square yards in order to provide enough grass for the horses old fence O What dimensions will require the least amount of fencing if no fencing is needed along the existing fence O grazing area O 300 yards by 300 yards 300 yards by 600 yards y 600 yards by 600 yards 200 yards by 900 yards

Determine the slant or nonlinear asymptote of f x Oy 7x 16 O y 7 O y 2 Oy 7x 2 7x 2x 9 X 2

Determine the local maximum and minimum values of f x 2x 3x 12x 4 using the second derivative test when it applies O O O O Local maximum value is 24 at x 2 Local minimum value is 3 at x 1 Local minimum value is 24 at x 2 Local maximum value is 3 at x 1 Local minimum value is 17 at x 1 Local maximum value is 8 at x 2 Local maximum value is 17 at x 1 Local minimum value is 8 at x 2

Identify all of the global and local extrema of the graph 6 5 4 3 2 1 6 3 0 3 4 6 4 is a global maximum of f x at x 3 O is a local maximum of f x at x 1 1 is a local minimum of f x at x 2 5 is a global minimum of f x at x 0 4 is a global and local maximum of f x at x 3 O is a local maximum of f x at x 1 1 is a local minimum of f x at x 2 5 is a global and local minimum of f x at x 0 4 is a global and local maximum of f x at x 3 O 5 is a global and local minimum of f x at x 0 4 is a global maximum of f x at x 3 O 5 is a global and local minimum of f x at x 0 3 4 5 6

Determine all the vertical asymptotes of f x O x 0 O x 2 Ox 0 x 3 x 4 Ox 0 x 3 2x 7x 4 x 7x 12x

Find all the critical numbers of f x x 8 and x 0 Ox 8 x 4 and x 0 Ox 4 and x 0 Ox 8 and x 4 3x2 x 4

Find the inflection point s for the function f x 2x 3x5 12x 2 O 0 2 and 1 11 O 0 12 and 1 15 O 1 19 and 0 2 O 0 0 and 1 0

Determine the intervals over which f x x 3x is increasing or decreasing O O O O f x is decreasing on 0 U 2 f x is increasing on 0 2 f x is decreasing on 2 U 0 f x is increasing on 2 0 f x is decreasing on 0 2 f x is increasing on 0 U 2 f x is decreasing on 2 U 2 0 f x is increasing on 0

Determine the interval s over which the graph of f x 2x6 6x5 17x 1 is concave up or concave down O O O O Concave up on 2 U 0 Concave down on 2 0 Concave up on 2 0 Concave down on 2 U 0 Concave up on 2 Concave down on 2 Concave up on 2 Concave down on 2

Approximate the area of the region bounded by y x 9 the x axis x 0 and x 4 by finding the combined area of the rectangles as shown in each figure and averaging the results O 13 5 units O 12 units2 O 17 units 0 9 2 2 7 4 1 0 9 2 7 4 1

Find the derivative of y log7 4x 25 O O dx dx dx dx 3 log7 4x 25 4x 25 24x log 4x 25 4x 25 24x log 4x 25 4x 25 In7 3 log7 4x 25 4x 25 In7

Identify all of the global and local extrema of the graph O O 6 5 4 3 2 1 5 4 3 0 43 4 5 6 2 O is a global maximum of f x at x 0 5 is a global minimum of f x at x 1 O is a global and local maximum of f x at x 0 0 8 is a local minimum of f x at x 2 5 is a global minimum of f x at x 1 3 O is a global and local maximum of f x at x 0 5 is a global minimum of f x at x 1 O is a global and local maximum of f x at x 0 5 is a global and local minimum of f x at x 1 4 5 6

Find the derivative of f x 2x 7 In O f x 6In x In 2x5 7 6 1 f x 24 5 7 O X O f x 6 X O f x 6 X 10x4 2x5 7 10x4 2x5 7 6

Calculate the slope of the line that passes through 12 5 and 2 9 O 7 5 01 17 7 7

Determine if the function f x x3 is differentiable at x 0 If not identify why O f x x is not differentiable at x 0 because lim x3 does not exist vant O f x x is differentiable at x 0 O f x x is not differentiable at x 0 because f 0 is not defined O f x x is not differentiable at x 0 because f 0 is not defined

The height h t in feet of a football thrown from a skyscraper after t seconds is given by h t 16t 73 3t 325 Find the velocity of the football after 7 seconds O 54 1 feet per second O 150 7 feet per second O 54 1 feet per second O 150 7 feet per second

A rancher has 200 feet of fencing to be used to enclose two adjacent congruent rectangular corrals y O 33 feet by 25 feet 25 feet by 50 feet O 33 X feet by 33 33 feet X What dimensions of the overall enclosure should be used so that the enclosed area will be maximized X y y

Find all the critical numbers of f x x4 4x 3x 2 then determine the local minimum and maximum points by using a graph O O O Critical numbers x 1 x 0 x 1 0 2 is a local and global minimum Critical numbers x 0 and x 1 0 2 is a local minimum Critical numbers x 1 and x 1 There is no local maximum or local minimum Critical number x 1 There is no local maximum or local minimum

Consider the following function 3x 4 x 2 3 f x 1 if x 2 if x 2 Determine if f x is continuous at x 2 If not select the option with the correct reasoning as to why not O Not continuous at x 2 because lim f x f 2 O Not continuous at x 2 because lim f x does not exist O Continuous at x 2 O Not continuous at x 2 because f 2 is undefined

Find the derivative of f x cot 2x4 10x O f x csc 8x 10 O f x 8x 10 csc 2x4 10x O f x csc 8x 10 O f x 8x 10 csc 2x4 10x

Which of the following is a function O 3x 4y 12 O 5x 5y 125 O 4y 7x x 63 O 15y 9x 27

Evaluate lim Olim possible then using direct substitution 11 X 2 X 2 X 2 Olim C X 2 X 2 X 2 Olim X 2 X 2 11 X 2 lim 1 1 X 2 X 2 X 2 1 X 0 does not exist algebraically by simplifying if 1 2 1 1 4

Find the derivative of f x O f x O f x O f x O f x 2cosx 9x 16 18x sinx 32sinx 54x cosx 9x 16 2sinx 27x COSX 3 18x sinx 32sinx 54x2cos 9x 16 18x3sinx 32sinx 54x cosx 9x 16

3 Determine if the function f x x is differentiable at x 0 If not identify why O f x x is differentiable at x 0 O O f x x is not differentiable at x 0 because lim x does not exist vant f x x is not differentiable at x 0 because f 0 is not defined f x x is not differentiable at x 0 because f 0 is not defined

Find the following indefinite integral x 2e 9x dx O O O 3 3 N W 2 2e 63x c 3 2e 9 C 9 3x 20 x c C

Consider the function f x x2 4x 5 Use it to find 2 f 3 and fa 4 2 7 O 3 8 f a 4 a 4a 15 f 2 7 O f 3 8 fla 4 a 40 5 f 2 7 Of 3 26 f a 4 a 4a 41 f 2 7 Of 3 2 fla 4 a 40 41

8 The total profit in thousands of dollars from selling x x fishing boats is given by P x 0 3x 1 A Find the total profit for selling 15 boats B What is the Marginal Profit for the 16th boat C What is the actual profit of selling the 16th boat

You are visiting your friend Fabio s house You find that as a joke he filled his swimming pool with Kool Aid which dissolved perfectly into the water However now that you want to swim you must remove all of the Kool Aid contaminated water The swimming pool is round with a 20 foot radius It is 7 feet tall and has 2 feet of water in it How much work is required to remove all of the water by pumping it over the side Use the physical definition of work and the fact that the weight of the Kool Aid contaminated water is a 63 3lbs ft

Find f g 2 for the following functions O wer to enter your answer opens in new window f x x 1 and g x 2x 5

If f x sec x and f a 6 find the exact value of the following a f a b f a f a 2x f a 4