Application of derivatives Questions and Answers

Use properties of logarithms to evaluate without using a calculator 2 log 4 log4 256 3 log 6561

Use graphing technology to find approximate solutions of log 6x 3 4 4 log 6x 3 help numbers X

7 pts Using Theorem 8 Maximum Minimum Principle 1 find the absolute maximum and absolute minimum values of the function if they exist over the indicated interval nobisnut gevie f x x 3x2 9x 7 on 2 2 allmos

The price p in dollars and the demand x for a particular clock radio are related by the equation x 5000 50p A Express the price p in terms of the demand x and find the domain of this function B Find the revenue R x from the sale of x clock radios What is the domain of R C Find the marginal revenue at a production level of 2800 clock radios D Interpret R 4400 76 00

Use power series operations to find the Taylor series at x 0 for the following function 13xe The Taylor series for ex is a commonly known series What is the Taylor series at x 0 for e 80 Type an exact answer n 0 Use power series operations and the Taylor series at x 0 for ex to find the Taylor series at x 0 for the given functio 8 Type an exact answer 1 n 0

Find the indicated quantities for y f x 2x f 4 Ax f 4 A Simplify Ax B What does the quantity in part A approach as Ax approaches 0 A f 4 Ax f 4 Ax 11

Find the following for the vectors v 10i 2k and u i j k a v u v u b the cosine of the angle between v and u c the scalar component of u in the direction of v d the vector proj u a v u 8 Type an exact answer using radicals as needed v 2 26 Type an exact answer using radicals as needed u 3 Type an exact answer using radicals as needed 4 b The cosine of the angle between v and u is 78 Type an exact answer using radicals as needed 4 26 c The scalar component of u in the direction of v is Type an exact answer using radicals as needed d proj u Type your answer in terms of i j and k Type an exact answer using radicals as

Find the following for the vectors v 2i 3j 6k and u 7i 24j a v u v lul b the cosine of the angle between v and u c the scalar component of u in the direction of v d the vector proj u a v u 86 Type an exact answer using radicals as needed v 7 Type an exact answer using radicals as needed u 25 Type an exact answer using radicals as needed 86 b The cosine of the angle between v and u is 175 Type an exact answer using radicals as needed 86 7 c The scalar component of u in the direction of v is Type an exact answer using radicals as needed d proj u Type your answer in terms of i j and k Type an exact answer using radicals as neec

Find the following for the vectors v 10i 2k and u i j k a v u v u b the cosine of the angle between v and u c the scalar component of u in the direction of v d the vector proj u a v u 8 Type an exact answer using radicals as needed v 2 26 Type an exact answer using radicals as needed u 3 Type an exact answer using radicals as needed 4 78 Type an exact answer using radicals as needed c The scalar component of u in the direction of v is Type an exact answer using radicals as needed b The cosine of the angle between v and u is

9 sin 0 4 cos 0 Replace the polar equation r The equivalent Cartestian equation is y 0 Choose the correct graph below OA with an equivalent Cartesian equation Then identify the graph B O C 10 O D

Find the polar coordinates 0 0 2n and r20 of the following points given in Cartesian coordinates a 1 3 b 2 2 2 2 c 2 3 2 a The polar coordinates of the point 1 3 are Q Type an ordered pair Type an exact answer using as needed Type any angles in radians between 0 and 2 b The polar coordinates of the point 2 2 2 2 are 0 Type an ordered pair Type an exact answer using as needed Type any angles in radians between 0 and 2 c The polar coordinates of the point 2 3 2 are 0 Type an ordered pair Type an exact answer using as needed Type any angles in radians between 0 and 2

Find parametric equations and a parameter interval for the motion of a particle starting at the point 3 0 and tracing the top half of the circle x y 9 two times Choose a correct possible parametrization below O A x 3 cost y 3 sint Ost 2 OB X 3 cost y 3 sint Ost 2 O c x 3 cost y 3 sint 0 t T OD x cost 3 y sin t 3 1 Ost 2

Given a function p x x x 2 x 2 A Explain why the Mean Value Theorem applies to p x on the interval 1 2 Find the point s guaranteed to exist by the Mean Value Theorem

Find the area enclosed by the given curves 12 y x y 4x A 8 B 16 C 2

Find the derivative S A 5x 6 14 dt 5t 6 B 5x 6 1 5 5x 6 2 C D 5 XE

Speedometer readings for a vehicle in motion at 3 second intervals are given in the table t sec 6 9 12 15 18 0 3 0 9 v ft s 31 56 71 68 50 Estimate the distance traveled by the vehicle during this 18 second period using a left sum L6 a right sum R6 and a middle sum M3 L6 705 R6 855 M3 474 Question Help Add Work feet feet x feet Video

Given the function graphed below evaluate the definite integrals 4 3 2 1 1 2 3 7 2 4 2 s f x dx f x dx Question Help Video Add Work 6 q

Estimate the areas under the curve by computing the left Riemann sum Lg 11 1 8 8

Evaluate the integral 16 22 dx by interpreting it in terms of area In other words draw a picture of the region the integral represents and find the area using geometry 16x dx 16 Holn Written Example

2 Next question The graph below shows a rectangular sum of n 4 rectangles to approximate the area under the line from x 0 to 2 4 14 13 12 11 10 9 8 7 6 5 4 3 2 Get a similar question You can retry this question below Is this a right hand or left hand sum left hand sum y What is the equation of the line What is the value of the sum 3 6 Complete the notation for the sum of the areas of all 4 rectangles

Consider a function h that is continuous over 10 10 and let 10 foto h 0 Use the above information to evaluate each of the following definite integrals 17 forth he 7 h x 7 dx 10 h x 7 dx Assuming h is odd Assuming h is even 10 10 h x dx h x dx 5 10 h x dx

Find the rate of change of total revenue cost and profit with respect to time Assume that R x and C x are in dollars R x 40x 0 5x C x 4x 20 when x 35 and dx dt 25 units per day The rate of change of total revenue is The rate of change of total cost is The rate of change of total profit is per day per day per day

circle is growing its radius increasing by 2 mm per second Find the rate at which the area is changing at the moment when the radius is 26 mm When the radius is 26 mm the area is changing at approximately Round to the nearest thousandth as needed second s mm mm per second second s per mm

Given the function f x x using 5 rectangles of equal width find an approximation to the area between the curve and the x axis over the interval 3 9 if the heights of the rectangles are found by evaluating the function at the left endpoints right endpoints and midpoints of each subinterval created by a partition of the interval given Using left endpoints the approximate area is Using right endpoints the approximate area is Using midpoints the approximate area is

1 Find a power series representation and interval of convergence for the derivative of 1 8 x f x

Find the partial derivatives fa and fy of the function f x y The variables are restricted to a domain on which the function is defined f x y 5x 3xy 3y fr x y 5x 3y 3y fy x y 5x 3x 9y X X

Given the functions f x x 6x g x 6x h x 5x 6 Evaluate the function h g x for x 6 Give your answer in exact form h g 6 is Undefined

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

Set up the form for the partial fraction decomposition Do not solve for A B C and so on 12t 8 2t 9t 18 00

6 3 Sx d A 1

5 Show that f x 2 e is an even function Sketch fand describe its symmetry 10 9 8 10

2y 2 y X dy 2y x 0 x 0 dx

3 Find the missing side lengths Leave your answers as radicals in simplest form 2 1 C x A x B x D x 5 n A m B m 30 C m 10 30 2 40 15 3 20 3 y 3 5 6 3 20 3 3 m 5 3 2 y 15 D m n 2 n 5 n 2 y 10 3 3 5 5 3 5 n 2 4 4 5 2 10 3 3 40 3 5 6 3 4 u 5 2 45 V A u 3 v B u 2 3 3 C u v 3 60 2 B a 3 2 3 a N W D u 3 v 2 V b A a 5 b 5 5 2 2 32 3 3 2 2 b 5 5 C a b 2 2 D a 10 b 10 5 2 2

Use the Law of Sines to solve the triangle If an answer does not exist enter DNE Round your answers to two decimal places A 119 a 125 b 202 B C C x X X

2 Given a function of g x 2x 8x 3 on 1 5 same as previous problem answer the following questions You cannot use the graphic calculator to find the answers A Use the First Derivative Test to identify the local extreme value of g x when x 2 ve Test to identify the local extreme value of g x when x 2

What two nonnegative real numbers with a sum of 23 have the largest possible product

56 59 Functions from derivatives Use the derivative f to determine the x coordinates of the local maxima and minima of f and the inter vals on which f is increasing or decreasing Sketch a possible graph of f f is not unique 56 f x x 1 x 2 x 4 10 sin 2x on 2m 2m 57 f x 58 f x x 1 x 2 x 3 59 f x x x 2 x 1

Given m x x 2 find mm x and write the domain in Interval notat Part 0 2 Part 1 of 2 mm x

2 Use the parent function f x bx and transformation techniques to graph h x 3 x 5 Also indicate which transformation s is are applied Determine the domain range and horizontal asymptote of h Select all transformations that apply A Reflect the graph of f about the x axis B Horizontally stretch the graph of f by a factor of 5 C Shift the graph of f 5 units to the right D Shift the graph of f 5 units to the left E Vertically stretch the graph of f by a factor of 5 F Vertically compress the graph of f by a factor of 5 G Reflect the graph of f about the y axis H Horizontally compress the graph of f by a factor of 5 1 Shift the graph of f 5 units down J Shift the graph of f 5 units up

Test the series for convergence or divergence Use the Select and evaluate lim 7140 Since the limit is Select Note Use INF for an infinite limit 7th n 10 2n 8 Select

Given the following graph what factor would appear in the denominator O x 1 O x 1 O x 0

5 Find where f x 0 Give your answer in interval notation O 2 infinity O 1 infinity O 2 1 O 2 1 Search PRE IC O

What is the multiplicity of the x 3 factor O Even O Not Enough Information Odd

Use the graph to find f x 0 Assume the function extends from negative infinity to infinity 2 O infinity 0 O infinity 0 u 0 3 O 3 infinity 3 infinity 10

10 The graph of a function f is shown where f x a z p z q for some constants a p and 9 Which of the following is true about the function f A a p q 0 B a 0 p q C D P 9 0

8 1 8e x 1 8e x dx

Let z 8 cos isin and W 3 3 cos Which statement describes the geometric construction of the product zw on the complex plane O Stretch z by a factor of 3 and rotateradians counterclockwise Stretch z by a factor of 3 and rotate radians counterclockwise O Stretch z by a factor of 24 and rotate radians counterclockwise O Stretch z by a factor of 24 and rotate radians counterclockwise sin 7 TE i sin

Find the Vertical Asymptote O x 1 O y 1 O x 1 O y 1 2 6

Use the end behavior to determine that as xx f x infinity 0 Onegative infinity 3

Question 11 Find the horizontal asymptote of f x y 4x 3x 1 4x32x 3