Application of derivatives Questions and Answers

5 Find the exact value 3 4 a b tan r c sin co 37 CO 2 d CSC 3

Sketch the region enclosed by the given curves y 3 0 y cos x y 2 cos x 0 x 2n 2 Find the area of the region Need Help Read It 3 2 2n y 3 2 00 1 KIN 2 BR 2 2x X 3 00 1 R 2 R 3x 2 2x X 10 1 RIN

Verify that wxy Wyx for w In 5x 8y Wxy 40 2 5x 8y

f 4 For the function f x y sin 9x 8y find dx of ax 11 and of dy

Vrite a in the form a T aNN without finding T and N a r t m sin t i m cost j ntk OT ON ENTENIA 117

Opposite vectors have O the same magnitude and direction the same direction the same magnitude different magnitudes and directions

For the following system a Determine how if at all the planes intersect 27 b If they do determine the intersection 3A 2x y 62 5 0 3x 2y 15z 9 0 5x x 3x y 1

Que Listen se the first principles definition of the derivative to find fil x x 37 11 wh

Listen M An airplane is flying at an airspeed of 400 km h on a heading of 035 A 175 wind is blowing from a bearing of 130 Determine the ground velocity of the airplane and the direction of flight Express the direction in quadrant bearing form 2K 3A

Listen n 3 4 Determine the scalar equation for the line with normal vector passing through the point P 1 2 1K 17 G 34

4 Listen Given the following graph of f x use your understanding of deriv a graph of f x 2C 24 2 3

E 4 Listen A manufacturing company is responsible for designing a can for dic which will have a volume of 500 mL Determine the dimensions of the can that will inimize surface area Recall SAcylinder 2rr 2mrh and Tr2h 2T 2A linder

Listen 200 1 Determine the slope of the normal to y 4 L III at the point 0 0 25 O GO 30

Listen Evaluate the following limit if it exists 1K 2A x 16 4 lim T 0

Examine the graph of the function below derivative positive x 0 3 T 1 24 The second derivative is never positive The second derivative is always positive 1 2 31 21 3 x

Listen Using the process shown in this unit sketch the function x 3x 9x 5 labelling all key points on your sketch Show all y x work to determine 3C 2A the y intercept any x intercept s Critical point s Inflection point s innals of increase decrease

Listen Determine the minimum value of the function f x 2 sin x cosx on the terval 0 n 1T 2A P of

Listen Given g x x 2x 1 the concavity on the interval 3 x 0 can described as ONeither concave up or concave down O Concave up Concave Down Unknown

Listen Determine the derivative of the following function Do not simplify 2A f x sin cos x RA

Graph the inequality 7x y2 4 Use the graphing tool to graph the inequality Click to enlarge graph 10 B B 10 8 6

Show all work Do not use a calculator 1 16 pts 1 Write the equation of the circle in standard form Then identify the cente graph

If g x x where the graph of fis shown in red evaluate g 3 y

For the function f x 7e find f x f 0 5 and f 5 f x Type an exact answer in simplified form

in dollars and the weekly sales x in thousands of units of a certain commodity satisfy the demand equation 8p x 187 500 Determine the rate at which sales are changing at a time when x 250 p 25 and the price is falling at the rate of 40 per week The rate at which sales are changing is thousand units per week

Assume x and y are functions of t Evaluate dy dt dx xy x 12 3 x 2 y 5 dt A 3 OB 9 O C 3 OD 9

A man who is 5 foot 9 inches tall is losing weight at the rate of 3 pounds per month Suppose that for a person of this height the relationship between body mass index B and weight w in pounds i given by 4600B 703w Find the rate of change of body mass index with respect to time Differentiate both sides of the equation with respect to time dw dt dB 4600 703 dt GID Find the rate of change of body mass index with respect to time Select the correct choice below and fill in the answer box to complete your choice Round to the nearest tenth as needed OA The man s body mass index decreasing by B The man s body mass index is increasing by BMI index points per month BMI index points per month

A company knows that unit cost C and unit revenue R from the production and sale of x units are related by C 254 000 changing by 14 and the revenue is 4 000 OA 399 05 OB 353 60 OC 444 50 On 5000 3536 Find the rate of change of revenue per unit when the cost per unit

The slope of the tangent line to a curve is given by f x 2x 9x 6 If the point 0 7 is on the curve find an equation of the ca f x

Divide 9x 21x 8x2 30x 5 by 3x 1

Find the real and complex zeros of the function f x x 13x 57x 85 Express your answer as a list separated by commas Do not use a sign in your answer instead list the zeros separately Provide your answer holour

Divide 45x 36x 15x 3x 16 by 5x 4

The graph of f is shown State the signs of f and f on the interval 0 2 f 0 f 0 2 X

Question Find the third degree polynomial function that has zeros 2 and 15i and a value of 1 170 when x 3 Provide your answer below

A farmer plans to enclose a rectangular pasture adjacent to a river see figure The pasture must contain 45 000 square meters in order to provide enough grass for the herd No fencing is needed along the river What dimensions will require the least amount of fencing X m y m

T x Find the tangent line approximation T to the graph of f at the given point Then complete the table Round your answer to four decimal places 0 1 2 4 f x X f x T x 3 9 3 99 4 4 01 MY NOTES 4 1 ASK YOUR TEACH

Use the graph of f to identify the critical numbers of f identify the open intervals on which f is increasing or decreasing and determine whether f has a relative maximum a relative minimum or neither at each critical number If an answer does not exist enter DNE a y X 2 2 decreasing 6 2 i Identify the critical numbers of f Enter your answers as a comma separated list 4 X ii Identify the open interval s on which f is increasing or decreasing Enter your answer using interval notation increasing relative maximum X iii Determine whether f has a relative maximum a relative minimum or neither at each critical number Enter your answers as a comma separated list relative minimum X

Points DETAILS se the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis y 5x x y 4 about x 1

Consider the following equation y 9ex x Find y x y x 9e 1 Find an equation of the tangent line to the curve at the point O y 1 9

Consider the following function f x x 8x a Find the critical numbers of f Enter your answers as a comma separated list X b Find the open intervals on which the function is increasing or decreasing Enter your answers using interval notatio an answer does not exist enter DNE increasing decreasing c Apply the First Derivative Test to identify the relative extremum If an answer does not exist enter DNE x y relative maximum relative minimum x y

d relative maximum 4 X 2 X decreasing i Identify the critical numbers of f Enter your answers as a comma separated list 2 4 ii Identify the open interval s on which f is increasing or decreasing Enter your answer using interval notation increasing relative maximum X iii Determine whether f has a relative maximum a relative minimum or neither at each critical number Enter your and a comma separated list relative minimum Untitled

Differentiate the function V t V t t 3 5 8 X

Consider the following Find f x f x f x x 4x 4x x4 Compare the graphs of f and f and use them to explain why your answer is reason f x 0 when f Select f x is positive when Select f x is negative when f Select

If g x xf x where f 2 8 and f 2 3 find an equation of the tangent line to the graph of g at the point where x 2 y

Consider the following equation y 9e x Find y x y x Find an equation of the tangent line to the curve at the point 0 9

Differentiate the function y y 2 x 8x 9x

c sin x cos x O sec x sec x tan x 1 cot x CSC x O cos x sin x 1 sin x O sec x O sec x tan x sin x cos x O cos x sin x cos x sin x tan x 1 sec x 1 cos x cot x 1 CSC x

Differentiate g t g t 9 7t 4t 3

Solve the equation and enter its solution set below If there is more than one solution enter the solutions in solution set notation separated by a comma and in order from smallest to largest If the solution set is the empty set enter it as Do not enter any spaces in the answer 3x 10 2 x

The perimeter of a rectangle is 20 feet Of all possible dimensions the maximum area is square feet when its length and width are both 5 feet Are there dimensions that yield a minimum area Yes the length would be 3 ft and width 7 ft Yes the length would be 1 ft and width 9 ft Yes the length would be 4 ft and width 6 ft No there are no dimensions that would produce an absolute minimum area

Sketch the region enclosed by the given curves y y 12x y x 6 8 6 2 4 X 0 6 6 6 DO 8 6 MY NOTES ASK YOUR TEACHEL