Application of derivatives Questions and Answers

Consider the function f x 2x 18x 42x 2 with This function has an absolute minimum at the point and an absolute maximum at the point 7 x 2

7x 5 angle of the angle 7x The reference angle for IS 5 Type your answer in radians Type an exact answer using x as needed Use integers or fractions for any the expression

Give the reference angle for angle measure 3 The reference angle is Simplify your answer Type an exact answer using as needed Use integers or fractions for any the expression

Choose the correct graph below OB 5 Select the correct choice below and if necessary fill in the answer box to complete your choice OA sin 0 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression OB The function is undefined

Select the correct choice below and if necessary fill in the answer box to complete your choice CODE OA sin 0 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression OB The function is undefined

Determine whether the Law of Sines or the Law of Cosines can be used to find another measure of the triangle B 11 cm 97 O Law of Sines O Law of Cosines 43 Then solve the triangle Round your answers to two decimal places b CI A

Click here to watch the video Sketch an angle 8 in standard position such that 0 has the least possible positive measure and the point 0 5 is on the terminal side of 0 Then find the values of the six trigonometric functions for the ang Rationalize denominators if applicable Do not use a calculator Choose the correct graph below A Ay Q Q OB F SUTHY Q 5 O C OU

Use the Law of Cosines to solve the triangle Round your answers to two decimal places C 106 a 10 b 7 A B

Graph the function over a one period interval y tan O A Which graph below shows one period of the function 1 I X 7 14 video 14 Q Q G OB Q ISA G C O C Q 76 17x 12 3 1 OD

Rewrite the function using the power reducing formulas f x 4 sin x f x X Use a graphing utility to graph the function y 1 5 1 0 0 5 1 5 1 00 5 2 2 2 2 0 5 1 0 0 5 1 5 1 0 1 5 1 5 1 0 0 5 1 5 1 0 0 5 4 y 2 2 y 2 2 0 5 1 0 0 5 1 0

1 y cscx CSC 1 a The period of y cscx is 2 Simplify your answer Type an exact answer using as needed Use integers or fractions for any numbers in the expression 1 b The phase shift of y csc x is 0 Type an exact answer using as needed Use integers or fractions for any numbers in the expression 1 c The range of y CSC X IS 2 Type your answer in interval notation Use integers or fractions for any numbers in the expression

y 6 csc zx www a The period of y 6 6 csc x is csc Type an exact answer using as needed Use integers or fractions for any numbers in the expression

Click here to watch the video Find the reference angle for the angle 223 The reference angle is Type an exact answer

Click here to watch the video Write tan 0 in terms of sec 0 if 0 is acute tan 0 expression

Use the graph of the function f shown to the right to estimate the indicated function values and limits Complete parts A through E E Is f continuous at x 3 Explain OA The function is continuous OB The function is discontinuous at x 3 since lim f x f 3 X 3 OC The function is discontinuous at x 3 since f 3 does not exist OD The function is discontinuous at x 3 since lim f x does not exist k Q Q G

The sum of the squares of two negative numbers is 117 and the difference of the squares of the numbers is 45 Find the umbers If there is more than one pair use the or button The pair s of numbers and 0 010 S

Perform the elementary row operation 3R R R on the given matrix Write numbers as integers or simplified fractions 3 1 4 14 2 1 2 3 Resulting matrix

Consider the function f x 4 2 8 on the interval 2 7 A Find the average or mean slope of the function on this interval Average Slope B By the Mean Value Theorem we know there exists at least one c in the open interval 2 7 such that f c is equal to this mean slope Find all values of that work and list them separated by commas in the box below List of values

The floor of a rectangular bedroom requires 210 ft of carpeting Molding is placed around the base of the floor except at two 3 ft doorways If 52 ft of molding is required around the base of the floor determine the dimensions of the floor 3 ft 3 ft

21 Given the velocity and initial position of a body moving along a coordinate line at time t find the position of the body as a function of time 8 V TO 4t sin 41 S 7 2 T A S 2 cos 4t 8 C s 2 cos 4 3 T B s 2 cos 41 4 TU 4t D s 2 cos 4 T 21

Refer to the functions f and g and evaluate the function for the given value of x Select Undefined if applicable f 6 5 5 8 6 3 2 2 4 6 and g 3 4 4 7 0 2 2 5 8 1 6 is f Undefined

the problem 20 Given the acceleration initial velocity and initial position of a body moving along a coordinate line at time t find the body s position at time t a 12 v 0 2 s 0 6 A s 61 2t 6 B s 12t 2t 6 C s 6t 2t 6 D S 6t 2t 20

ex dx 1 e2x A sin 1 ex C C 2 1 e2x C B sec 1 ex D ex sin 1 ex

Find all critical numbers for the function and then list them separated by commas in the box below List of critical numbers Hint There are three critical numbers f x x 13 x 1

Consider the function f x 5x 4x on the interval 5 5 A Find the average or mean slope of the function on this interval Average Slope B By the Mean Value Theorem we know there exists at least one c in the open interval 5 5 such that f c is equal to this mean slope Find all values c that work and list them separated by commas in the box below List of values

Consider the function f x 2 8x on the interval 2 8 A Find the average or mean slope of the function on this interval i e 8 f 2 8 2 B By the Mean Value Theorem we know there exists a c in the open interval 2 8 such that f c is equal to this mean slope For this problem there one c that works Find it C

By applying Rolle s theorem check whether it is possible that the function f x x x 12 has two real roots Final answer input possible or impossible Answer this part last after you have throught through the reasons below Your reason is that if f x has two real roots then by Rolle s theorem f x must be input a number here at a certain value of x between these two roots but f x is always input negative positive or zero

roblem 2 Find an expression for the area under the graph of f x 112 6 on the interval 5 10 D lim 81x A lim 72 00 G lim n 5i n 11 5 1 6 5 5 i 1 11 5 SWI SWI WI Problem 2 Colect i 1 i 1 10 n 5 10i n 11 5 0 6 5 n 5 10i 5 n n 11 5 0i 6 n B lim n x 10i H lim n n E lim n n n 5i 5 n 11 5 5 6 n 5 5i n 5i n 11 i 6 n n C lim n co 11 5 6 n 10 n 10i n 11 5 0i 6 n 5 n F lim t 10 11 10 6 i 1 10i n

Let F x 1 f 3 dt Find F 4

Problem 3 The graph off is given to the right 1 Evaluate f x dx y f x 3 2 1 3 2 1 L 2 d 2

The graph of a function f is shown Which graph is an antiderivative of f DOO 0 2 y 2 0 1 5 0 0 5 0 5 1 0 b 0 2 0 4 0 6 0 8 X

A stone is dropped from the upper observation deck of a tower 400 m above the ground Assume g 9 8 m s a Find the distance in meters of the stone above ground at time t s t b How long does it take the stone to reach the ground Round your answer to two decimal places S c with what velocity does it strike the ground Round your answer to one decimal place m s d If the stone is thrown downward with a velocity of 5 m s how long does it take to reach the ground Round your answer to two decimal places

A company estimates that the marginal cost in dollars per item of producing x items is 1 67 0 002x If the cost of produci one item is 552 find the cost in dollars of producing 100 items Round your answer to two decimal places

Find f f t f t 2 cos t sec t n 2 t 2 f 3 3

Find f f t 3 f 4 11 f 4

A box with a square base and open top must have a volume of 13 500 cm Find the dimensions of the box in cm that minimiz the amount of material used sides of base height cm cm

A manufacturer has been selling 1 000 flat screen TVs a week at 500 each A market survey indicates that for each 10 rebate offered to the buyer the number of TVs sold will increase by 100 per week a Find the demand function price p as a function of units sold x p x b How large a rebate in dollars should the company offer the buyer in order to maximize its revenue c If its weekly cost function is C x 64 000 120x how should the manufacturer set the size of the rebate in dollars in order to maximize its profit

If you are offered one slice from a round pizza in other words a sector of a circle and the slice must have a perimeter of 36 inches what diameter pizza in inches will reward you with the largest slice in

Find the area of the largest rectangle that can be inscribed in the ellipse 1

at d its to point pposite A B the other side of the lake in the shortest possible time see the figure She can walk at the rate of 4 mi h and row a boat at 2 mi h For what value of the angle shown in the figure will she minimize her travel time G 11 A 0 2 B 2 C

At time t the position of a body moving along the s axis is s t 9t 24t m a Find the body s acceleration each time the velocity is zero b Find the body s speed each time the acceleration is zero c Find the total distance traveled by the body from t 0 to t 3 a What is the body s acceleration each time the velocity is zero Select the correct choice below and if necessary fill in the answer box to complete your choice OA m s2 Simplify your answer Use a comma to separate answers as needed OB There is no solution b What is the body s speed each time the acceleration is zero Select the correct choice below and if necessary fill in the answer box to complete your choice OA m s Simplify your answer Use a comma to separate answers as needed OB There is no solution c What is the total distance traveled by the body from t 0 to t 3 m Simplify your answer CL

X Simplify your answer Type an exact answer using radicals as needed

The figure below is a contour diagram of f x y In each of the following cases list the marked points in the diagram there may be none or more than one at which a 0 b f 0 C C c 0 O ogress 0

Find the critical point s of f x y f x y x 3y 12x 6y 3 f x y has Choose one critical point s

Given nonzero vectors u v and w use dot product and cross product notation as appropriate to describe the following a The vector projection of u onto v b A vector orthogonal to u and v c A vector orthogonal to u xv and w d The volume of the parallelepiped determined by u v and w e A vector orthogonal to u xv and uxw f A vector of length u in the direction of v B k uxv x uxw C kvxw OD k uxv xw f Which of the following describes a vector of length u in the direction of v OA V ul OB lul Oc lulv A

Given honzero vectors u v and w use dot product and cross product notation as appropriate to describe the following a The vector projection of u onto v b A vector orthogonal to u and v c A vector orthogonal to u xv and w d The volume of the parallelepiped determined by u v and w e A vector orthogonal to u xv and uxw f A vector of length u in the direction of v d Which of the following describes the volume of the parallelepiped determined by u v and w A vxu w B uxv w C D e If k is any constant which of the following describes a vector orthogonal to u xv and uxw u v xw uxv w OA k uxv uxw B k uxv x uxw OC Kvxw OP kluxv xw

Find the length and direction when defined of u xv and vxu u 8i 3j 5k v 4i 4j 2k luxv 6 17 Type an exact answer using radicals as needed Select the correct choice and fill in any answer boxes in your choice below A The direction of u x v is 14 i 4 Type exact answers using radicals as needed OB The direction of u xv is undefined 20 k

a Find the area of the triangle determined by the points P Q and R b Find a unit vector perpendicular to plane PQR P 1 1 0 Q 0 0 1 R 0 0 1 a The area of the triangle is Type an exact answer using radicals as needed

Expand log 34 to rewrite as a sum difference or product of logarithms Choose all correct answers There may be more than one way to rewrite the expression A log a log b c4 OB log a log b log c OC 2 log a 3 log b 4 log c OD 2 log a 3 log b 4 log c OE log a log b c4 OF None of the above

Expand In expression A In 4 In k OB kln 4 OC In 1 k ln 4 OD In 1 In 4 OE In 1 In 4 to rewrite as a sum difference or product of logarithms Choose all correct answers There may be more than one way to rewrite the OF None of the above