Application of derivatives Questions and Answers

53 cos 8 csc 8 5 and 8 E Q

1 12 13 and 14 Verify that the function satisfies the three hypotheses of Rolle s heorem on the given interval Find all numbers c that satisfy the conclusion of olle s Theorem 11 f x 2x 4x 5 1 3 12 f x x 2x 4x 2 2 2

Is the graphed function differentiable at x 1 Choose the correct answer below OA Yes the function is differentiable at x 1 OB No because while the function is continuous at x 1 it does not have a tangent line at x 1 OC No because the function is not defined at x 1 OD No because while the function is defined at x 1 and lim exists lim is not equal to the value of the function at x 1 X 1 X 1 OE No because while the function is continuous at x 1 it has a vertical tangent line at x 1 OF No because while the function is defined at x 1 lim does not exist X 1 10 8 r 4 2 6 y 2

The given function is defined for all x except for one value of x If possible define f x at the exceptional point in a way that makes f x continuous for all 8 x 64 f x X 0 Select the correct choice below and if necessary fill in the answer box to complete your choice OA f x 8 x 64 X for x 0 for x 0 OB There is no range value that will make f x continuous at x 0

4 Ex 3 Graph f x x y Name the asymptotes Domain Range X X Y

The graph shows an idealized description of the temperature in F for approximately the last 150 thousand years of a location at the same latitude Complete parts a c www a Find the highest and lowest temperatures recorded The highest temperature is The lowest temperature is F F Average Annual Temperature Idealized 140 000 105 000 70 000 Years ago 80 70 60 150 40 30 35 000 10 F

The function graphed is of the form y a sin bx or y a cos bx where b 0 Determine the equation of the graph y Use integers or fractions for any numbers in the expression 6 Ay 6

Use the given graph to estimate the value of each derivative a f 3 b 1 2 c f 1 d f 0 e f 1 1 f 2 9 f 3 y f x Sketch the graph of f 2 7 N ya 3 y 3 2 3 4 3

Write and simplify but do not evaluate an integral with respect to x that gives the length of the following curve on the given interval y 5 cos 3x on TR 2 4 An integral that gives the arc length is Tuno exact answers 4 I RIN 5 cos 3x dx

Among all rectangles that have a perimeter of 64 find the dimensions of the one whose area is largest Write your answers as fractions reduced to lowest terms

17 If cos 0 V3 2 and terminates in QI find sin 0

13 Find cose if sin and terminates in QII

Use the given graph to estimate the value of each derivative a f 3 b f 2 c f 1 d f 0 e f 1 f f 2 g f 3 y f x YA 1 0

6 For 60 90 determine if each statement is true or false V4 2 a b sin 0 0 cos 0 c tan 0 1 3 2 1 3

5 If is an angle in the first quadrant and cos 0 find in radians and degrees

4 Use the unit circle to estimate the values of sin 55 and cos 55 to the nearest tenth 0 5 55 O 0 5 0 05 0 5

1 Consider the angles 0 where 0 0 2 a What is the largest possible value of cos 0 At which angle measure 0 does it occur b What is the smallest possible value of cos 0 At which angle measure 0 does it occur c What is the largest possible value of sin 0 At which angle measure 0 does it occur d What is the smallest possible value of sin 0 At which angle measure 0 does it occur

Suppose f x 0 for all x and the y axis A Set up the integral that gives the volume of the solid Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers 4 S 2x f x dx 0 OB S x dx The volume of the solid is Type an exact answer 0 f x dx 15 Let R be the region in the first quadrant bounded by the coordinates axes y f x and x 4 Find the volume of the solid generated by revolving R aroun CORA

Use the shell method to find the volume of the solid formed when a hole of radius 6 is drilled symmetrically along the axis of a right circular cone of radius 8 and height 12 Model the situation on a set of axes by placing the center of the base of the cone at the origin and the cone s axis along the positive y axis Set up the integral that gives the volume of the solid using the shell method Use nonnegative and increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers OA S OB dy dx

Use a graphing utility to graph the function and to approximate any relative minimum or relative maximum values of the functio Round your answers to two decimal places If an answer does not exist enter DNE f x 5x 4x 9 relative minimum relative maximum x y x y

onsider the following f x 2x a Use a graphing utility to graph the function y increasing 10 10 constant decreasing 5 5 10 5 5 10 y 10 5 5 10 5 5 10 10 X X 10 10 5 5 y 10 5 5 10 y 10 5 5 10 b Determine the open intervals on which the function is increasing decreasing or constant Enter your answers using interval notation If an answer does not exist enter DNE 5 5

Evaluate the function at each specified value of the independent variable and simplify If an answer is undefined enter UNDEFINED f x a b 1 3x f 5 f 0 25 X 0 X 0 c f 5 II

Antarctica is roughly semicircular with a radius of 2 00 x 103 km see the figure The average thickness of its ice cover is 3 00 x 10 m How many cubic centimeters of ice does Antarctica contain Ignore the curvature of Earth 8000 m 2000 km

Determine the slope m and y intercept if possible of the linear equation If the slope is undefined enter UNDEFINED Enter NONE if there is no y intercept 5x 7y 1 slope m y intercept x y Describe its graph The line is vertical and passes through 2 0 O The line is horizontal and passes through 0 O The line passes through 0 1 and falls 7 units for each horizontal increase of 5 units The line passes through 0 1 and falls 5 units for each horizontal increase of 7 units O The line passes through 0 1 and falls units for each horizontal increase of units

3 For each function find the inverse the domain and range of the function domain and range of the inverse and determine whether the inverse is a functi a f x x 7 b f x 2x 2 c f x x d g x 2x 1 3

Question pod area between f x and the x axis over the interval

Find the volume of the solid generated when the region bounded by y x and y 2 x is revolved about the x axis Set up the integral that gives the volume of the solid 10 dx 0 CHILD

Find the net signed area between the curve of the function f x 2x 4 and the x axis over the interval 7 3 Do not include any units in your answer

T T C 2 For f x sin x and g x 0 5x on the interval 2 2 instantaneous rate of change of fis greater than the instantaneous rate of change of g for which value of x A 0 8 D 1 2 B 0 E 1 5 C 0 9 the

Tafu is working with subatomic particles in the Physics lab A positron is traveling in a straight line down the particle accelerator Tafu models its position with the function 2 p t t 1 where t 1 In the limit calculations below if you need to use or enter INFINITY or INFINITY a Calculate the limit lim p 0 11 b Calculate the limit lim p t 1 c Calculate the limit limp 1 1 2 d Let a 1 be some fixed number Let h 0 be some small number Write out a formula for the slope of the secant line to the graph of x p t between the point at t a and the point at t ath You don t need to simplify this formula yet WARNING When you write out an expression involving a and h anytime you want to multiply a times h you need to leave a single space between the two l e a h means a times h In contrast If you write ah the computer webassign thinks that you are talking about a new variable called ah Alternatively use the arithmetic buttons in the pop up box whenever you want to add subtract multiply or divide e Keep a fixed and think of your formula in d as a function in the variable h Call it f h Now simplify your formula for f h so that it is a fraction with polynomials in the numerator and denominator and the denominator does not vanish at h 0 L e your final answer should be a rational function in h which is defined at h 0 WARNING When you write out an expression involving a and h anytime you want to multiply a times h you need to leave a single space between the two I e a h means a times h In contrast if you write ah the computer webassign thinks that you are talking about a new variable called ah Alternatively use the arithmetic buttons in the pop up box whenever you want to add subtract multiply or divide 1 Calculate the limit lim h h 9 A time t I when the positron s Instantaneous velocity is zero is

Worksheet d f x 7x 12x 20 e f x f f x 3e 8x 4 x 2 Math 124

C8 y 2x cos x in the interval 0 x 5 is A 6 B 7 The number of times the second derivative is zero for the graph of D 9 E 10 C 8

Worksheet Math 124 Worksheet for Week 4 Limits and Derivatives This worksheet reviews limits and the definition of the derivative with graphs and computations 1 Answer the following questions using the graph y f x below The function f x has domain all numbers except 7 as seen from the graph a lim f x b lim f x 2 7 c f 0 d lim f x e lim f x 2 0 T f lim h 0 f 3 h 5 h g f 5 HDD 11 30 41 h lim h o i lim h 0 f 8 h h f 8 h f 8 h f 6 h f 6 h i o k lim h 0 Week 4 f 3 h 5 h 1 List all the intervals where the derivative f x is negative m List all the intervals where the derivative f x is decreasing n A critical value for f x is any r in the doma of f x where f x 0 or f x is undefine List all critical values of f x

3 A Consider the graphs of y 3x c and y 6x where c is a real constant Determine all values of c for which the graphs intersect in two distinct points

CURRENT OBJECTIVE Find the total distance traveled by a particle given its velocity function Question Find the total distance traveled by a particle according to the velocity function v t 3t 9 m sec over the time interval 2 10 Enter your answer as an exact fraction if necessary and do not include units in your answer

particle given its velocity function Question Given that a particle travels according to the velocity function v t 2t 8 m sec what is the total distance traveled by the particle over the time interval 2 8 Enter your answer as an exact fraction if necessary and do not include units in your answer

12 Show that the vectors 1 2 6 3 2 1 and 7 10 9 lie on the same KU 3 plane

Find the volume of the solid generated in the following situation The region R bounded by the graph of y 2 sin x and the x axis on 0 x is revolved about the line y 2 The volume of the solid generated when R is revolved about the line y 2 is Type an exact answer using as needed cubic units

Let R be the region bounded by the following curves Find the volume of the solid generated when R is revolved about the x axis Recall that cos x 1 2 y cos 2x y 0 x 0 The volume of the region revolved about the x axis is cubic units Type an exact answer cos2x

2 Consider the following utility function defined over two goods 1 and 2 U x1 x2 x x2 The prices of goods 1 and 2 are p and p2 respectively a Does the law of diminishing marginal utility hold for good 2 Find the MRS of good 1 for good 2 15 b Write the equation representing the budget constraint assuming the consumer s income is M 3 c Using the method of Lagrange maximize the utility subject to the budget constraint What are the demand functions for and 2 Assuming p 2 P2 1 and M 100 find the quantities of x and x2 that maximizes utility 25 d Check the SOC for the Lagrangian method 12

A graphing calculator is recommended 5x 1 x a If F x F 2 y x find F 2 and use it to find an equation of the tangent line to the curve y 10 b Illustrate part a by graphing the curve and the tangent line on the same screen y 10 10 5 44 X 10 10 5 5 5 10 O 10 y 10 5x 1 x2 F10 at the point 2 2 5 10 DO 10 y 10 10 5 y 10 5 10 X O

The displacement in feet of a particle moving in a straight line is given by s s 9 16 where t is measured in seconds a Find the average velocity in ft s over each time Interval 1 4 8 6 8 8 10 v 8 12 10 10 ft s b Find the instantaneous velocity in ft s when t 8 ft s 20 P s c Draw the graph of s as a function of t and draw the secant lines whose slopes are the average velocities in part a Then select the graph of s with the tangent line whose slope is the instantaneous velocity in part b 0 30 ft s PU S 10 10 10 15 NE 5 10 153 20 00 DO 10 15 10 15 10 10 20 0 30 10

Two different suppliers are competing to supply a particular part for a new Boeing airliner The Japanese supplier charges J n dollars for n parts whereas the Italian supplier charges I n dollars for n parts Keep in mind that the people in Boeing would like to minimize their costs a Which supplier should be used to supply a large number of parts if lim xIn n I n b Which supplier should be used to supply a large number of parts if J n n lim QJn O I n c Which supplier should be used to supply a large number of parts if J n lim x I n 0 lim 74X n O n c Which supplier should be used to supply a large number of parts if J n I n 0 2 QJn I n 5

Find the vertical asymptote x A f x x 3 5x 4

The velocity function below corresponds to the motion of a particle If the velocity is measured in meters per second what is the net displacement of the particle for the first 5 seconds S 6t 2 if 0 t 2 if 2 t 5 4t 2 Enter your answer as an exact fraction if necessary and do not include any units Provide your answer below v t

5 Determine the y intercept of a line with a slope of 2 that is a tangent to the curve y y x 2 4x 5

Question An amusement park has a marginal cost function C x 1000e 5 where x represents the number of tickets sold an a marginal revenue function given by R x 60 0 1x Assuming that the park has no fixed costs find the total profit generated when selling 550 tickets Round your answer to the nearest dollar

Question Determine the area in square units of the region bounded by g x 10x 11 and f x x 5x 13 over the interva 8 3 Provide your answer below

Suppose that a particle moves along a straight line with velocity defined by v t 2t 2 where 0 t 4 in meters per second Find the total distance traveled up to t 4 Provide your answer below

7 The graph shown here is the graph of which of the following equations 3 a y 2x 3 b y x 2 c y x 2 5 4 3 2 5 4 3 2 1 1 3 y TS 5 1 345