Application of derivatives Questions and Answers

Refer to the figure to the right Set up a definite integral from a to b that represents the shaded area I Ay b y kx

Find the derivative of the function g x g x e 5x 8 5x 3 2 ez 3 52

Use the Fundamental Theorem of Calculus to find the derivative of 1 f x 1 t5 dt f 5 f x

Evaluate the limit using L H pital s rule e 3x 1 3x lim 0

2 10 4x dx 1

Find the area between y x 2 and y x 4 Round your limits of integration and answer to 2 decimal places F

Find the area of the region enclosed by f x x and g x V Write an exact answer fraction Answer

Determine the area enclosed by f x x 2x 6x 7 and g x x 3x 8x 17 The region in question lies between x The upper function is g x The area is A and x and the lower function is f x

Find the area between y x 2 and y x 4 Round your limits of integration and answer to 2 decimal places The area between the curves is square units

Find the area between y 9 and y x 1 with x 0 a The area between the curves is square units

H1 10 9 7 6 5 4 3 y A x 2 1 y H x 8 3 2 1 27 1 2 1 2 3 4 5 6 7 8 9 10 Write an integral that represents the area of the shaded region shown above Use the function names of th curves given da

sin x Recall that tan x cos x Using integration by substitution to derive the derivative of tan x Edit

S 5 sin 9x 1 dx a This can be transformed into a basic integral by letting Consider the indefinite integral u 9x 1 and du 9dx Jamai b Performing the substitution yields the integral cos 9x 1 C one 9 X syntax error Check your variables you might be using an incom c Once we integrate and substitute the final answer in terms of x is 9 cos 9x 1 C

Determine if the given polar coordinates represent the same point 8 1 2 8 3x 2 DO Yes No

Find a vector of magnitude 3 in the direction of v 12i 5k The vector is i 0 Simplify your answer Use integers or fractions for any numbers in the expression

Find the length and direction when defined of u xv and vxu u 6i 4j 6k v 3i 2j 3k OA i k Type exact answers OB The direction of uxv is not defined lux v Simplify your answer What is the direction of uxv Select the correct choice below and if necessary fill in the answer boxes to complete your choce of 6 A i k Type exact answers vxu Simplify your answer What is the direction of vxu Select the correct choice below and if necessary fil in the answer boxes to complete your ch ction of vxu is not defined

Consider the indefinite integral This can be transformed into a basic integral by letting u ln z and du 1 Z In 25 5 ln z dz Z dz Performing the substitution yields the integral In C du syntax error Check your variables you might be using an

Compute the given integral sin sin r cos xdr

Evaluate the indefinite integral 2 x 14 x4dx

Use the shell method to find the volume of the solid generated by revolving the shaded region about the x axis The volume is This question 1 point s possible 3 Submit y 2 V2 0 X

Compute the given integral 5x 1 5x 2x 3 dx 40

Find 6 csc 2 da CS fo 6 csc 22 d dx

Find the volume of the solid generated by revolving the shaded region about the y axis The volume of the solid generated by revolving the shaded region about the y axis is 1 0 5 x 2 tan 2 5

Consider the indefinite integral In z dz In 2 Z This can be transformed into a basic integral by letting u du and dz Performing the substitution yields the integral du

Evaluate the following indefinite integral S 3 3t 3 dt

of 11 cos 6x dx Find S 11 cos 6x dx 11 sin 6x 6 a similar question You can Hint Did you include the constant of integration Question Help Written Example Add Work X

5 sin 9x 1 dx a This can be transformed into a basic integral by letting Consider the indefinite integral u du and b Performing the substitution yields the integral c Once we integrate and substitute the final answer in terms of x is Question Heln Writton Eva

What is the solution set to log x 7 log3x O 0 3 O 7 0 0 3 3 00 1012 FOTO RIGH

The slope of the tangent line to a curve is given by f x 4x 7x 8 If the point 0 7 is on the curve find an equation of the curve f x

F x y z cos z i 3ysin 3z j 6x zk a Find div F b Find curl F

Solve the initial value problem dy 2e sin e 25 y in 5 0 5 The solution is VE

Functions f x and g x are composed to form h x x 2 If f x x 2 and g x x a what is the value of a 0

For the complex number z 150 210 O 240 330 21 2 T 2 i what is arg z

7 5 points Which of the following is the right hand sum with n 2 rectangles for f x sin z on interval 2 x 4 A 2 sin 2 sin 0 sin 2 sin 2 B 2 sin 0 sin 2 sin 4 2 sin C 3 sin 2 sin 1 3 sin in 1 D 3 sin 1 sin 4 in 4 sin 4 Esin 1 sin 4

4 5 points Which of the following is a true statement about the function f x graphed below 4 3 2 4 2 Y f b 3 X A on the interval 20 the function f x has a global minimum at z 3 and a global maximum at z 2 B on the interval z 20 the function f x has a global maximum at x 2 and a 0 global minimum at C on the interval z 20 the function f x has no global minimum and no global maximum D on the interval z 20 the function f x has a global maximum at x 2 and no global minimum

5 5 points What is Ar for either a left or right hand sum with n 4 rectangles for 34 2h A Az 8 5 B Az 4 C Az 8 D Az 34 E Ax 32 f x da

2 5 points A function f t has a local minimum at t 2 and a local maximum at t 4 Assuming t 2 and t 4 are critical points found by setting f t 0 and solving for t which of the following must be true Min t 2 210 max t 4 9 t 0 Af 2 is positive and f 4 is positive B f 2 is positive and f 4 is negative C f 2 is negative and f 4 is negative D f 2 is negative and f 4 is positive 4 420 242 0

Question 2 Solve this system of equations 4y x 4x No solution 4y 17 20

3 Solve this system of equations 4x 3y x y 29 2

lesha wants to prove that the measures of the 3 angles in a triangle always have a sum of 180 She decided to prove this conjecture using rigid motion She began with triangle ABC shown in Figure 1 B 16 Figure 1 C Figure 2 Rotate triangle ABC degrees on the midpoint of Rotate triangle ABC degrees on the midpoint of ince we used 2 rigid motions and all three angles form a straight line and a straight line equals angles will equal 180 degrees we can show that all

11 Approximate da using a Riemann sum with n 3 terms and the midpoint rule What does this number have to do with In 4 12 The velocity of a particle is given by v t 1 1 m s a Find the acceleration a 2 of the particle W How for does the particle travel in the first 5 seconds

A ball is thrown into the air by a baby alien on a planet in the system of Alpha Centauri with a velocity of 46 ft s Its height in feet after t seconds is given by y 46t 17t A Find the average velocity for the time period beginning when t 1 and lasting for the following number of seconds 01 sec y 10 1 y 1 0 01 X syntax incomplete Hint If the throw starts at t 1 and lasts 01 sec what will the ending time t2 be 005 sec 45 1 X 002 sec 46 4 001 sec 45 83 X NOTE For the above answers you may have to enter 6 or 7 significant digits if you are using a calculator Estimate the instanteneous velocity when t 1 12 X

8 sec 0 tan 0 3

Compute the derivatives of the functions arctanx a y 2arctana In2 2 1 x 1 x 4

2 BUSINESS REVENUE If a company s marginal revenue function is MR z xe 4 find the revenue functic Hint Evaluate the constant C so that revenue is 0 at x 0 X 4 xe dv U EX du 1 dx 4 4 x du e v e

se integration by parts to find the integral x ln x dx C

3 Express 7 45 as a ratio of integers

Given the graph below solve the following inequality h x 20 10 1 x 9 8 7 30 9 8 7 6 5 3 2 1 2 6 8 9 10 6 7 8 9 10

Que Find the radian measure that corresponds to the given degree measure 225 a 40500m Obl 54 Od 5

Find the value of COS given that sin a 4 6 11 b 4 6 5 6 TT and ta