Calculus Questions

Use this definition with right endpoints to find an expression for the area under the graph of f as a limit Do not evaluate the limi f x 4 sin x 0 x A lim 816 w1

Use the Midpoint Rule with n 4 to approximate the integral 2 7x MA 7x 9 dx

Let f x 9 4x a Estimate the area under the graph of f the x axis and the lines x 1 and x 2 using three rectangles and a right endpoints R Improve your estimate by using six rectangles R6 Sketch the curve and the approximating rectangles for R y 0 1 25 20 15 10 0 1 25 20 15 H 10 5 0 1 25 20 15 10

Oil leaked from a tank at a rate of r t liters per hour The rate decreased as time passed and values of the rate at two hour time intervals are shown in the table t h r t L h 8 8 7 5 6 9 6 2 5 6 Find lower and upper estimates for the total amount of oil in liters that leaked from the tank over the interval 0 10 Use five equal subintervals lower estimate upper estimate 0 L L 2 10 5 1

3g f x 2 cos x 0 sxs 4 evaluate the left Riemann sum with n 6 taking the sample points to be left endpoints Round your answer to six decimal places x What does the Riemann sum represent The Riemann sum represents the net area of the rectangles with respect to the x axis Illustrate with a diagram y 0 24 100 744 3x 8 151 00 24 15100 3 RIN tisto 8 3x 4 08 2 00 4 2 3 15 00 5x 3x

The graph of the velocity of a braking car is shown v ft s 90 60 30 0 2 4 6 t seconds Use this graph and a midpoint sum with six equal subintervals to estimate the distance traveled in feet by the car while the brakes are ft

1 Prove the following cos2x 1 sin2x cotx 2 Find all solution in 0 27 Sin 2x sin x 0

Rewrite the equation in standard form then tell the positive b value of the following 9y 4x 18y 32x 91 0 4 9 01 156 None of these 3 12 5 6 2

Determine if the following equations represent a parabola an ellipse or a hyperbola x 1 4 y 3 45 100 49x 25y 294x100y 884 0 y 12y x 28 0 9x2 16y2 36x 96y 36 0 x 1 y 2 16 25 y 2 20 x 2 25 1 1 1 Parabola 2 Hyperbola 3 Ellipse

Find the exact values of the center vertices and foci of the following equation 2 3 v 2 1 16 Choose the one best and most complete match for each Foci Vertices Center 1 3 2 2 2 3 3 3 2 4 1 2 and 7 2 5 0 2 and 6 2 6 3 1 and 3 5 7 3 2 and 3 6 3 2 7 3 7 2 8 9

Determine the correct graph for the following equation y 8y 4x 8 0

Determine the correct graph for the following equation x 3 y 2 16 9 ERCORSERIE 1

Determine the correct graph for the following equation x 1 12 y 3

Solve for x in the interval 0 2 Give exact values where appropriate If an exact answer cannot be found give a decimal answer rounded to the nearest 2 decimal places 3 cos x 5 cos x 4 If there is more than one answer give your answers separated by a comma with no spaces in between For example 7 89 3 70

Use Stokes theorem to evaluate F dr F x y z zexi z 6y j x 9z k C is the circle y2 z 9 x 4 oriented clockwise as viewed from

Find the average value of the function f p 4 0 p over the solid ball p 9 The average value is 27 4 Simplify your answer Type an exact answer using x as needed

Find the gradient field of the function f x y z x y 3z 1 2 The gradient field is Vf 1 x y 3z xi yi 3zk

Find and dx dy 2 2 f x y 4x y 4x dx 4x y ax y ay 4x y Type an exact answer using radicals as needed Type an exact answer using radicals as needed

2 Find the average distance from a point P x y in the disk x y 9 to the origin The average distance is 2 Simplify your answer

Find the maximum value of s xy yz xz where x y z 15 The maximum value is www

a The curvature of the helix r t a cost i a sin t j bt k a b 0 is k 2 a b The largest value x can have for a given value of b is What is the largest value x can have for a given value of ba

Find the line integral along the given path C f x y dx where C x t y 9t 2 for 0 st 7 x y dx

x 10x 12 x x 2 3x 3 Determine the correct partial fraction decomposition for the expression above Enter your answer as the sum of fractions with a constant in each numerator and with a linear expression in each denominator Provide your answer below

Find the curl of the vector field F F 5xy z i 3yz x j 2xz y k curl F i k www

The result is calculated as follows 1 x 7 10 dx 107x17 10 1 X 0 F 1 F 0 0

Therefore an antiderivative of f x x7 10 is found as follows x17 10 F x x17 10

Consider the following graph YA B y 2xy The function in the graph is a solution of one of the following differential equations A y 1 xy 0 C y 1 2xy Decide which is the correct equation and justify your answer For points in the first quadrant the differential equation y 1 xy Select in the first quadrant and Select Select For points in the first quadrant the differential equation y 2xy Select in the first quadrant and Select Select For points in the first quadrant the differential equation y 1 2xy Select in the first quadrant and Select Select To summarize the graph is the solution to which of the three differential equations OA y 1 xy OB y 2xy Oc y 1 2xy Need Help Read It V Since the function grap For points in the second quadrant this differential equation Select in the second quadrant this differential equation Select be the correct equation V Since the function graph For points in the second quadrant this differential equation Select in the second quadrant this differential equation Select be the correct equation For points in the second quadrant this differential equation Select in the second quadrant this differential equation Select be the correct equation Since the function gra

1 Consider the data below for a car tire with a leak Minutes after the leak began Pressure of air in the tire in kilopascals kPa 0 400 5 335 10 295 15 20 255 225 25 195 30 170 a Calculate the average rate of change over the 30 minute interval Explain the meaning of this rate using proper units

2 For the function f x x 3x 2 Use the graph of the function to estimate the instantaneous rate of change at x 2 by drawing a tangent line and calculating it s slope

3 Evaluate each limit if it exist a x 8x lim X 3 2x 1

4 What is the fundamental difference between average rate of change and instantaneous rate of change Use the concept of tangent and secant lines to explain

Find an equation of the curve that passes through the point 0 6 and whose slope at x y is y 36

Write out the form of the partial fraction decomposition of the function as in this example Do not determine the numerical values of the coefficients a x5 4 x x x 4x 4 b A B r 1 x x x 6 B A x 3 x 2 Cx E x 2 X Dx F x 2 X

f f t f t dt where f is the function whose graph is shown Let g x 3 g 5 g 6 2 1 2 3 a Evaluate g x for x 0 1 2 3 4 5 and 6 g 0 g 1 g 2 g 3 g 4 f 4 5 6 7 t

The table below shows the number of accidents fatalities hours flown and miles flown for a country s airlines with scheduled service in selected years Complete parts a and b below Miles flown billions 7 2 7 5 7 7 Year 2005 2008 2012 Accidents Fatalities 53 87 37 22 23 1 Hours Flown millions 17 1 17 5 17 3 SCITE a Compute the accident rate per million flight hours in 2005 and 2012 By this measure has general aviation become safer Select the correct choice below and fill in the answer boxes to complete your choice Round to the nearest tenth as needed OA General aviation has not become safer because the accident rate in 2005 hours OB General aviation has become safer because the accident rate in 2005 accidents per million flight hours is lower than the accident rate in 2012 accidents per million flight hours is higher than the accident rate in 2012 accidents per million flight accidents per million flight hours

Finding the Projection of u onto v In Exercises 37 38 39 40 41 42 43 and 44 a find the projection of u onto v and b find the vector component of u orthogonal to v 37 u 6 7 v 1 4 Answer 38 u 9 7 v 1 3 39 u 2i 3j v 5i j Answer 40 u 2i 3j v 3i 2j 41 u 0 3 3 v 1 1 1 Answer 42 u 8 2 0 v 2 1 1 43 u 9i 2j 4k v 4j 4k

Finding the Angle Between Two Vectors In Exercises 11 12 13 14 15 16 17 and 18 find the angle between the vectors a in radians and b in degrees 11 u 1 1 v 2 2 Answer 12 u 3 1 v 2 1 13 u 3i j v 2i 4j Answer 14 u cos 17 15 u 1 1 1 v 2 1 1 Answer i sin 2 14 j v 16 u 3i 2j k v 2i 3j 3 3 cos 377 1 sin 37 J 4 2 121

Finding Dot Products In Exercises 3 4 5 6 7 8 2 and 10 find a u v b u u c v d u v v and e u 3v 3 u 3 4 v 1 5

A graphing calculator is recommended A hawk flying at 19 m s at an altitude of 228 m accidentally drops its prey The parabolic trajectory of the falling prey is described by the equation x 57 y 228 until it hits the ground where y is its height above the ground and x is its horizontal distance traveled in meters Calculate the distance traveled in m by the prey from the time it is dropped until the time it hits the ground Express your answer correct to the n tenth of a meter 185265 004 Xm

Evaluate the definite integral 1 2 6 21 dr 1 2

A graphing calculator is recommended Find the general indefinite integral Use C for the constant of integration cos x 2x dx Illustrate by graphing several members of the family on the same screen 10 5 y 20 15 10 C 10 C 5 C 0 10 5 10 y 15 10 10 5 f 1 10 C 11

Evaluate the integral R 6 S sec 0 tan 0 de

Evaluate the integral 0 2 cos 0 de

An antiderivative of x as long as n 1 is

olution of the differential equation that satisfies the given initial condition dA Ab cos br A 0 b dr b sin br In 6 e No REVIOUS ANSWERS SCALCET9 9 3 015

Comparing Vectors In Exercises 21 22 23 24 25 and 26 determine whether u and v are orthogonal parallel or neither 21 u 4 3 v 1 2 3 Answer 22 u i 2j v 2i 4j 23 u j 6k v i 2j k Answer 24 u 2i 3j k v 2i j k 25 u 2 3 1 v 1 1 1

Alternative Form of Dot Product In Exercises 19 and 20 use the alternative form of the dot product to find u v 19 u 8 v 5 and the angle between u and v is 3

the angle between the vectors a in radians and b in degrees 11 u 1 1 v 2 2

y t 50e t 30 In 2 50 2 50 Exercise b How much of the sample remains after 40 years 50 2 Step 1 After 40 years we have the following y 40 50 2 mg mg Round your answer to two decimal places

1 0 1 Points DETAILS A T Find the area of the region that lies inside the first curve and outside the second curve r 3 cos 8 r 1 cos 8 PREVIOUS ANSWERS x Enhanced Feedback Please try again keeping in mind that you should first find the intersection points Then use the formula for the area between two parametric curves which is A Watch it Ion con dl where 2