Application of derivatives Questions and Answers

35 Bike Tykes is a company that makes bikes for children The company s weekly marginal profit function is given by P x 30x 0 3x 250 dollars per bike when x bikes are sold each week If the company s profit from selling 9 bikes each week is 0 i e the break even quantity is 9 bikes find the company s profit when 80 bikes are sold each week

Use a triple integral to find the volume of the solid bounded below by the cone 2 z x y and bounded above by the sphere x y z 288 2 The volume of the solid is 2304 2 3456 cubic units 0 0 288 X 2 y 2 288 z x y

4 Determine whether the following statements are true and give an explanation or counterexample a When using the shell method the axis of the cylindrical shells is parallel to the axis of revolution b If a region is revolved about the y axis then the shell method must be used c If a region is revolved about the x axis then in principle it is possible to use the disk washer method and integrate with respect to x or the shell method and integrate with respect y a Choose the correct answer below OA False The shell method produces shells that are always perpendicular to the axis of revolution True The shell method always produces shells parallel to the y axis True The rotation of the function about the axis creates a cylindrical shape parallel to the axis OD False The rotation of the function about the axis creates a disk shape perpendicular to the axis B c b Choose the correct answer below OA True Any time a function is revolved about an axis the shell method must be used OB False The shell method can only be used when a region is revolved about the x axis C True Then the axis of the cylindrical shells is parallel to the y axis thod may be used by integration with respect to y

the region bounded by the following curves Use the shell method to find the volume of the solid generated when R is revolved about the y axis y 13 x y 0 x 1 and x 10 Set up the integral that gives the volume of the solid using the shell method Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers O A O B 1 dy CED dx Sa

Estimate the area under the graph of f x over the interval 1 4 using ten approximating rectangles and right endpoints as sample points Rn Ln Repeat the approximation using left endpoints as sample points x 2 Repeat the approximation using midpoint as sample points M

Identify the integral Choose the correct answer below OA 2 f O C b 2n f x OD a b OB 2 g x f x dx a nethod that equals the volume of the resulting solid Tegloh bounded by the graphs of f and g and the lines x a and x b is revolved about the y axis Write the integra b 2x f x g x dx b a 2xx g x f x dx 2nx f x g x dx a

The price p in dollars and demand x for wireless headphones are related by x 8 000 0 15p The current price of 95 is decreasing at a rate 5 per week Find the associated revenue function R p and the rate of change in dollars per week of revenue R p

A manufacturer of handcrafted wine racks has determined that the cost to produce x units per month is given by C 0 4x 9 000 How fast is the cost per month changing when production is changing at the rate of 15 units pe month and the production level is 80 units Costs are at the rate of Round to the nearest dollar as needed per month at this production level

A retail store estimates that weekly sales s and weekly advertising costs x both in dollars are related by 40 000 20 000 0 00025x The current weekly advertising costs are 4 000 and these costs are increasing at a rate of 600 per week Find the current rate of change of sales The current rate of change of sales is Round to the nearest dollar as needed per week

A 13 foot ladder is placed against a vertical wall Suppose the bottom of the ladder slides away from the wall at a constant rate of 2 feet per second How fast is the top of the ladder sliding down the wall when the bottom is 12 fee from the wall The ladder is sliding down the wall at a rate of Type an integer or a simplified fraction RYYB ft sec

A boat is being pulled toward a dock as shown in the figure to the right If the rope is being pulled in at the rate of 4 feet per second how fast is the distance between the dock and the boat decreasing when it is 3 feet from the dock The distance between the dock and the boat is decreasing by dock Round to the nearest thousandth as needed Boat Rope 4 ft Dock feet per second when the boat is 3 feet from the

A ODEV f x IX olsun f x f n 1

A note card company has found that the marginal cost per card of producing x note cards is given by the function below where C x is the marginal cost in cents per card Find the total cost of producing 780 cards disregarding any fixed costs C x 0 05x 90 for x 1000 The total cost is cents

Suppose that for a company manufacturing calculators the cost revenue and profit equations are given by C 90 000 30x R 300x 20 where the production output in 1 week is x calculators If production is increasing at a rate of 600 calculators per week when production output is 5 000 calculators Find the rate of increase decrease in cost revenue and profit A Costs are Simplify your answer P R C at the rate of S per week at this production level

Solve Check for any extraneous solutions 3x 19 x 5 O x 1 only x 2 and x 3 x 6 only x 1 and x 6

Part III Circle the correct answer for each proble Be sure to fill out the Multiple Choice Answer Page at the tan x h tan x 7 Find lim A tan s C sec x tan x B sec x D see x 8 At which of the following values of x is g x defined but g x is undefined L x 2 II x 2 III x 3 A I and II only B I and III only C II only D II and III only Graph of g x

5 3 O 2 34 a The graph of a function y f x is shown Make a sketch of the antiderivative F given that F is continuous and F 0 1 7 5 20 2 m 6 2 5 6 7 8 Clear All Draw

1 2 c If k x g x AM 2 1 2 9 x 5 2 sl a x 4 2 MIN find k 1 3 points 1 6 The functions f x and g x are differentiable for all values of x Selected values for Cry and g x and their derivatives are shown in the chart above a If s x f x g x find 2 1 point b If h x f x g x find 1 2 points 4 d If f x e f x find 1 Leave your answer in terms of e 2 points

alues are given in is permitted 17 Consider the function f x x x 11 a Find the average rate of change of f x on 0 6 2 points b Find the instantaneous rate of change of f x when x 4 2 points c Write an equation of the line tangent to fat the point where x 4 2 points Still using f x above if g x f x find all values of x where the tangent lines to g am Horizontal Show how you arrive at your answer 3 points

Next question X Suppose f x 12x Determine and graph four distinct antiderivatives of f 10 8 6 4 2 Clear All Draw 10 8 6 4 2 2 4 6 8 Get a similar question YOU 10 2 4 6 8 16 retry COSES

9 The graph of a function g is shown It is known that g has only real zeros a Identify the degree of g b How many turning points does g have c State the zeros of g and their multiplicity d Write the equation for g x in intercept form e Find one other ordered pair on the graph of g that is neither an z nor a y intercept 2 41 104 8 7 6 0 0 2

The answer above is NOT correct A farmer must plant a rectangular garden comprised of subsections in a grid pattern 10 long and 10 wide Additionally she must bury irrigation pipes around the perimeter and between each of the sections If she has 1254 meters of irrigation pipes what is the maximum of the whole garden

7 Given the following graph of f x there are local extrema at xX X3 and xs There are points of inflection at x an X4 State the intervals in which the following conditions hold X 4 X 31 O y f x X X 1 a f x 0 f x 0 b f x 0 f x 0 8 If f x x ax b has a turning point at 2 3 determine the values of a and b

Find the area between the curves f x 4x 16x in dark blue and g x 3x 12x in dark red over the interval 1 5 The area between the curves is NOTE The graphs of the functions given intersect on the given interval a

THE CHAIN RULE If g is differentiable at x and f is differentiable at g x then the composite function F f g defined by F x f g x is differentiable at x and F is given by the product F x f g x g x In Leibniz notation if y f u and u g x are both differentiable functions the dy dy du dx du dx Note addition formula to execute next assignment d dx ax ax In a 2 y 4 3x 2 4 y tan sin x cin of

9 Determine if each statement is true or false Explain and justify your answer a If f 2 0 then f x as a local maximum or a local minimum when x 2 b If f x is a polynomial of degree 3 then its graph has one possible point of inflection

2 Which values for x below are solutions to the inequality 5 s 2 a 10 b 20 c 2 2 d 6 e 9 x NX

4 For the function f x x 4 x a State the domain cp pat 491 04 00 00 194

4 Find an inverse for the following functions Be careful Some of this func tions are not injective so determine a maximal domain where the inverse is defined a f x 3x 2 d k x x 3 x 1 b g x e m x c h x x 4 21 f n x x4 5 when x 0 x 3

3x 4x 5x

For f x 9x 9 and g x x find the following composite functions and state the domain of each a fog b gof c fof d gog ww a fog x 9x 9 Simplify your answer Select the correct choice below and if necessary fill in the answer box within your choice OA The domain of fog is x Type an inequality Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed B The domain of f o g is all real numbers b gof x Simplify your answer

For f x x and g x x find fog x and gof x Then determine whether fog x gof x What is fog x fog x x What is gof x 0 gof x

For f x x and g x x find fog x and gof x Then determine whether fog x gof x What is fog x fog x

Find the Maclaurin polynomials of orders n 0 1 2 3 and 4 and then find the nth Maclaurin polynomials for the function in sigma notation Enter the Maclaurin polynomials below for Po x P1 x P2 x P3 x P4 x Pn x m 0

ritz Benjamin buys a car costing 15500 He agrees to make payments at the end of each monthly period for 5 years He pays 4 8 interest compounded monthly What is the amount of ach payment Find the total amount of interest Fritz will pay Fritz s monthly payment is Round to the nearest cent Fritz will pay a total of Round to the nearest cent in interes

Find the monthly house payments necessary to amortize a 4 8 loan of 205 700 over 15 years The payment size is Round to the nearest cent

Suppose that f x and g x are given by the power series f x 7 6x 4x 5x and g x 6 9x 5x 2x By multiplying power series find the first few terms of the series for the product h x f x g x co C x c x 3x Co 11 C1 11 C2

5 Let A x f t dt with f x shown in the graph below 4 3 P 1 2 3 4 5 a sim 3 question You can retry this question below Add Work 5 At what a value s does A x have a local max x 2 9 Use commas to separate multiple values At what a value s does A x have a local min a value This is not a decimal or integer value Use commas to separate multiple values Restrict your responses to values 0 x 6 Q 0 9 4 9 X X This is not a decimal or integer

The graph below shows the constant function y c t 8 7 6 5 4 3 2 8 7 Answer 5 A 3 2 1 1 2 3 45 5 6 18 1 2 3 4 5 6 7 8 A new function w is defined as an integral w x c t dt 1 What is the value of w 3 We know that w is an antiderivative of c What is the formula for w

3 A motorcycle accelerates at a constant rate from 0 km h to 105 km h in 18 0 seconds

Match the sequence with its graph 4n n 1 an 10 8 6 4 2 an 10 8 6 4 N 2 4 2 4 8 10 6 8 10 an 10 8 16 4 an 10 8 4 2 2 4 6 8 6 8 10

Simplify the factorial expression n 3 n

Write the first five terms of the sequence defined recursively 1 2ak a a2 a3 a4 a5 a 80 ak 1

Write an expression for the apparent nth term an of the sequence Assume that n begins with 1 1 6 11 16 21 an

Find the indicated term of the sequence a4 an n 3n 1

2 The graph of which of the following quadratics does NOT have a vertex on the x axis 1x 1x A y 3 x 2 x 2 Ich B 5 x 3 yx x 16 Dy x 4 x 4 8 4 6 Za 1 2 6 X 4 y 4 86 48 16

Is your estimate an underestimate or an overestimate underestimate overestimate b Repeat part a using left endpoints L4 4 Sketch the graph and the rectangles y y 4 M 3 A 8 R 4 f x 4 cos x 3 8 f x 4 cos x 8 E X 4 S f x 4 cos x 3x 8 f x 4 cos x X

a Estimate the area under the graph of f x 4 cos x from x 0 to x x 2 using four approximating rectangles and right endpoints Round your answers to four decimal places R4 Sketch the graph and the rectangles y 4 R 8 R 4 f x 4 cos x WebAssign Plot 3x 8 f x 4 cos x X A A 4 f x 4 cos x 3 8 f x 4 cos x

question 13 Hand draw the Bode Diagram both magnitude and phase plots for the following transfer function Show all intermediate steps s 1 s 10 s 100 H s 100

Name Instructions Use algebraic techniques to solve problems unless otherwise instructed Show the calculations leading to your answers except basic arithmetic Simplify and circle your final answers 1 6 pts What is the domain of the function f x log X 3 7 x Show the work sign analysis 2 8 pts a What is the average rate of change of f x log x between 1 9 and 27 Write the answer as a fraction no decimals or mixed numbers b What is the equation of the secant line through 1 9 f 1 9 and 27 f 27 Use fractions in the