Calculus Questions

Simplify 6 6 20 6 3 62 56 24 b 5 Written Example

and direction What is the velocity vector 6 j 4t k is the position of a particle in space at time t Find the particle s velocity and acceleration vectors Then write the particle s velocity at t 0 as a product v t 3 i 16t j 4 k What is the acceleration vector a t 0 i 16 j 0 k Write the velocity vector at t 0 as a product of the speed and direction v 0 5 3 0 4 k Type exact answers using radicals as needed

The path r t t i 21 1 j describes motion on the parabola y 2x 1 Find the particle s velocity and acceleration vectors at t 3 and sketch them as vectors on the curve The velocity vector at t 3 is v 3 1 12 J Simplify your answer including any radicals Use integers or fractions for any numbers in the expression The acceleration vector at t 3 is a 3 0 i 4 j Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Choose the correct sketch for the particle s velocity and acceleration at t 3 O A OB 2 Q O C

Evaluate the function at each specified value of the independent variable and simplify If an answer is undefined enter UNDEFINED f x x 6 a f 8 14 b f 8 2 c f t UNDEFINED

The equation r t 3t 1 i 81 6 j 4t k is the position of a particle in space at time t Find the particle s velocity and acceleration vectors Then write the particle s velocity at t 0 as a product of its speed and direction What is the velocity vector v t 3 i 16t j 4 k What is the acceleration vector a t i j k

The tangent line to a smooth curve r t f t i g t j h t k at t to is the line that passes through the point f to g to h to parallel to v to the curve s velocity vector at to Use r to and r to to find parametric equations for the line that is tangent to the given curve at the given parameter value t to r t In X 1 3 1 m K j tln t 5 What is the standard parametrization for the tangent line y Z k to 3

Suppose lim f x 17 and f x is nonconstant x 5 Provided the limit exists 7 12 1 f x 10 lim x 5 f x 17 Question Help Video A Written Example

ne position of a particle in the xy plane at time t is r t t 5 i t 4 j Find an equation in x and y whose graph is the path of the particle Then find the particle s velocity and acceleration vectors at t The equation for the path of the particle is y x 10x 29 The velocity vector at t 4 is v Simplify your answers UCLEB

Use a graphing utility to graph the function and estimate its domain and range Then find the domain and range algebraically Enter your answers using interval notation h t 25 2 5 5 5 0 domain range X

Given lim f x 8 and lim g x 7 evaluate x 3 I 3 lim 6 f x 3g x x 3 lim 6 3f x I 3 lim 2 3 off 6f x 3g x 6 3f x 2 lim 6f x 3g x 6 3f x z 3

8 x x if x 4 if x 4 2x 3 Calculate the following limits Enter DNE if the limit does not exist lim f x Let f x lim f x 2 4 lim f x 2 4 Question Help Video B Written Example

Find the domain of the function Enter your answer using interval notation g x 4 7x 0

Suppose f x 7x 5 6 Evaluate lim f x x 5

Use the graph of the function to find the domain and range of f Enter your answer using interval notatic domain range b 5 4 3 c 0 00 9 f 3 f 0 X d f 4 X 10 y 987654321 7 140 5 2 2 1 1 IIIII 1 123456789 2 Use the graph to find the indicated function values a f 4 3 5 9L 1 2 y f x 4 5 X

Find the limit x 8x 15 lim 3 3x 91 10 Written Example

Evaluate the limit 4x 16 4 x 12x 32 lim Video 2 Writ

Suppose you know that lim f x 16 x 9 Find the following 16 f x lim x 9 f x 4 FA

Evaluate the limit lim y 0 1 8 y Y 1 64 11

Find the following values for f x sin I f x Based on these values lim sin I 0 7 0 0 0181 Olim sin 2 0 7 0 73354562801787 Olim sin 1 0 7 Be sure your computing device is set to RADIAN mod does not exist 0 00191 0 000103

Given lim f x 8 and lim g x 4 evaluate x 3 x 3 lim g x ln 6f x 5 x 3 lim 69 2 e f a lim 6f x 5g x x 3 lim x 3 cos f x g x

Suppose lim f x 8 and lim g x 3 Find lim f x g x x 7 x 1 x 1 I 1

Suppose lim f x 8 and lim g x 6 Find 24 7 24 7 lim 6 x 7 x 3g x 3g x 4x 4

Estimate the limit numerically or state that the limit does not exist 9 1 lim 0 X Give your answer to at least three decimal places xample

Find the following values for f x sin X f x Be sure your computing device is set to RADIAN mode 0 1 0 01 0 001 Based on the values it appears lim f x I 0

Estimate the limit numerically or state that the limit does not exist 4 sin 9x lim 0 sin 8x Be sure your calculator is set in RADIAN mode and give your answer to at least three decimal pla Question Help Video Written Example

The equation r t cos 2t i sin 2t j t20 describes the motion of a particle moving along the unit circle Answer the following questions about the behavior of the particle a Does the particle have constant speed If so what is its constant speed b Is the particle s acceleration vector always orthogonal to its velocity vector c Does the particle move clockwise or counterclockwise around the circle d Does the particle begin at the point 1 0 a Select the correct choice below and fill in any answer boxes within your choice OA The particle s constant speed is OB The particle s speed is not constant

The position of a particle in the xy plane at time t is r t t 5 i 1 2 4 j Find an equation in x and y whose graph is the path of the particle Then find the particle s velocity and acceleration vectors at t 4 The equation for the path of the particle is y

The equation r t 3t 1 8t 6 j 4t k is the position of a particle in space at time t Find the particle s velocity and acceleration vectors Then write the particle s velocity at t 0 as a product of its speed and direction What is the velocity vector v t i j k GEL

Problems 27 34 refer to right triangle ABC with C 90 In each case solve for all the missing parts using the given information 27 A 20 c 24 in 28 A 40 c 36 in 29 B 31 a 76 c 5 8 mi 35 m b 97 m 33 b 150 cm c 200 cm 30 B 21 c 4 2 mi 32 a 78 m b 83 m 34 b 320 cm c 650 cm

The tangent line to a smooth curve r t f t i g t j h t k at t to is the line that passes through the point f to 9 to h to parallel to v to the curve s velocity vector at to User to and r to to parametric equations for the line that is tangent to the given curve at the given parameter value t to r t cos t i What is the standard parametrization for the tangent line X y 12 5 sin t j k to 0 Z

The tangent line to a smooth curve r t f t i g t j h t k at t t is the line that passes through the point f to g to h to parallel to v to the curve s velocity vector at to Use r to and r to to find parametric equations for the line that is tangent to the given curve at the given parameter value t to r t 1 in 6 tim x k What is the standard parametrization for the tangent line X y Z to 3 F Tuno exprossions using t as the variable CIDE

The path r t t i 31 4 j describes motion on the parabola y 3x 4 Find the particle s velocity and acceleration vectors at t 1 and sketch them as vectors on the curve The velocity vector at t 1 is v 1 i j Simplify your answer including any radicals Use integers or fractions for any numbers in the expression

The equation r t 2t i 2t 16t j is the position of a particle in space at time t Find the angle between the velocity and acceleration vectors at time t 0 The angle between the velocity and acceleration vectors at time t 0 is radians Type an exact answer using as needed

volume of a shifted solid of revolution Question Define R as the region bounded above by the graph of f x x 2 below by the x axis on the left by x 1 and on the right by x 2 What is the volume of the solid obtained by rotating R about the line y 1 Enter an exact value in terms of

1 point Evaluate the integral x x 2 x dx

1 point Evaluate the integral 1 2 1 6 5 csc t cot t dt

Question Consider the region bounded by the graphs of f x 0 8e0 6x g x 10 x 4 3 and Select the correct answer below x 2 If this region is rotated about the x axis what is the volume of the resulting solid 2 0T13 O 3 2 07 Off 0 8e0 6x 3 0 64e 2x 2 0 O 3 2 10 x 4 O 2 100 T L 0 8e0 6x dx 10 x 4 X 2 100 x 4 0 64e 2x dx dx dx

1 point Evaluate the integral dt cos 71 1 tan 7t

1 Particle A moves along a horizontal line with a velocity where is a positive continuous function of t The time t is measured in seconds and the velocity is measured in cm sec The velocity of the particle at selected times is given in the table below t sec v cm sec 0 1 7 2 6 8 5 7 7 4 15 6 Use data from the table to approximate the distance traveled by particle A over the interval 0 ts10 seconds by using a right Riemann sum with four subintervals Show the computations that lead to your answer and indicate units of measure 2 1 3 4 5 2 417 3 f x 141 7cm 2 Let y t represent the population of a town over a 20 year period where y is a differentiable function of t The table below shows the population recorded at selected times t yrs 4 10 13 20 y t people 2500 2724 3108 3697 4283 10 24 9 Use data from the table to find an approximation for y 12 and explain the meaning of y 12 in terms of the population of the town Show the computations that lead to your answer y 12 y 13 y 10 13 10 3697 3108 13 10 Use data from the table and a trapezoidal approximation with four subintervals to approximate the average population of the town over the 20 year period Show the antations that lead to your answer F

6 Given the graph of f x find f x dx

Life is not about finding yourself Life is about creating yourself Lolly Daskal The rate at which water flows into a tank in gallons per hour is given by a positive continuous function R of time t The table below shows the rate at selected values oft for a 12 hour period 3 t hrs R gal hr 0 2 6 8 10 12 12 5 13 4 13 9 14 3 14 6 14 8 14 7 4 Use a midpoint Reimann sum with 3 subintervals to approximate the amount of water that flowed into the tank over the 12 hour period Use your calculator for 4 5 Show all work that leads to your answer 4 The total area enclosed between the curve y x 3x x 2 and the x axis on the interval 2 2 G 4 1 5 Find the average value of the function f x x 3x 8 on the interval 1 4 6 Given the graph of f x find f x dx

CURRENT OBJECTIVE Use the washer method to find the volume of a solid of revolution around the y axis Question Find the volume of a solid of revolution formed by rotating the region bounded on the right by the graph of f y y 2 and on the left by the graph of g y over the y axis interval 1 4 about the y axis Enter an exact value in terms of J Provide your answer below

if x 30 y 50 and 1 1 if the slope of the linear regression equation is 0 4 Then the estimated value of y at x 8 is Select one 0 b 4 6 d 6

Use the graph of y f x to graph the function y f 4 x 1 Label the points on the sketch For each point on the graph of f x compute the location of the corresponding point on the graph of y f 4 x 1 Point on f x 3 4 Point on f 4 x 1 Type an ordered pair 10 10 16 f x

blem 1 20 Points Use the graph of each function to identify its domain and its range N f x x 2 f x 4x 3 5 0 12 10 7 2

a For what values of x is the function f x 1x 251 differentiable Enter your answer using interval notation X X Find a formula for f f x b Sketch the graph of f 20 10 40 X 20 X f x 30 20 Sketch the graph of f 10 10 30 if x if x f x 40 20 20 40 10 20 20 X X 40 10 40 f x 20 WM 10 10 10 10 f x 40 20 20 40 20 40 20 f x 10 10 10 f x 20 10 10 20 5 10 10 20 X 20 f x 30 VAV 10 20 10 10 10 20 30 f x 20 10 10 20 10 10 20 20 9

Problems 19 26 refer to right triangle ABC with C 90 Write all answers that are angles to the nearest tenth of a degree 19 If A 40 and c 16 cm find a 20 If A 20 and c 42 cm find a 21 If B 20 and b 12 ft find a 22 If B 20 and a 35 ft find b

Find the first and second derivat y x 8 y y 1 2 x 8

wire 18 meters long is to be cut into two pieces One ece will be shaped as a square and the other piece ill be shaped as a circle see the figure Answer arts a c 18 m 4x 18 4x O a Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the square A x Simplify your answer b What is the domain of A domain Type interval notation c Graph A A x For what value of x is A smallest meters Use the MINIMUM function Round to two decimal places

Let P be a point on the graph of y x The tangent line at P will intersect the graph at the point Q a If P has x coordinate a then what is the equation of the tangent line at P y x b The line in a will intersect the curve y x in two places one of these is P and the other point is Hint This part involves factoring a cubic in x In general this may be difficult However in this case since we know point P is an intersection point x a must be a solution so x a must be a factor You should now be able to f c The ratio of the slope of the curve at Q to the slope at P is