Calculus Questions

CURRENT OBJECTIVE Use substitution to evaluate a definite integral with the power rule Question Find d 3x 3x 2 dx Enter your answer as an exact fraction if necessary

CURRENT OBJECTIVE Use substitution to evaluate a definite integral with the power rule Question Evaluate the following definite integral 5x x 1 dx Enter your answer as an exact fraction if necessary

Find the volume of the solid generated by revolving the region bounded by y 2x y 0 and x 3 about the x axis Set up the integral that gives the volume of the solid S 0 dx GEER

CURRENT OBJECTIVE Use substitution to find an indefinite integral with the power rule Question Find the following antiderivative 7x 3x 2 dx Do not include the constant C your answer For example if you found the antiderivative was 2x C you would enter 2x

Question Find x x 1 dx Submit your answer in fractional form Provide your answer below

The displacement in feet of a particle moving in a straight line is given by s 12 9t 16 where t is measured in seconds a Find the average velocity in ft s over each time interval 1 4 8 3 6 8 2 8 10 0 iv 8 12 20 10 10 ft s b Find the instantaneous velocity in ft s when t 8 1 ft 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 tar 20 ft s 10 EM 10 15 10 15 20 30 ft s X ft s Do 10 15 10 5 5 10 15 20

The solutions x y of the equation x2 16y2 16 form an ellipse as pictured below Consider the point P as pictured with x coordinate 2 a The ellipse is not given by a single function as it fails the vertical line test It is given by two function The points P and Q shown are in the upper part which can be described as y f x were f x b Let h be a small non zero number and form the point Q with x coordinate 2 h as pictured Then the y coordinate of Q is given by c Let h be a small non zero number and form the point Qwith x coordinate 2 h as pictured The slope of the secant line through PQ denoted s h is given by the formula d Rationalize the numerator of your formula in c and simplify to rewrite the expression so that it looks like f h g h subject to these two conditions 1 the numerator f h defines a line of slope 1 2 the function f h g h is defined for h 0 When you bu finished your simplification neither the numerator nor the denominator should become zero when h 0 When you do this 1 h g h e The slope of the tangent line to the ellipse at the point P is lims h 110

Question 4x Include the constant C in your answer Evaluate 4x 1 2x x 3 4 dx Provide your answer below

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

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 c f 0 d lim f x e lim f x x 0 x f lim h 0 15 14 13 12 11 101 761 g f 5 f 3 h 5 h 25 20 15 10 3 10 h lim h 0 j lim h 0 10 11 12 13 14 k lim h 0 f 8 h f 8 h f 8 h h f 6 h f 6 h 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 z in the domain of f x where f x 0 or f x is undefined List all critical values of f x

Worksheet Math 124 2 Evaluate the following limits and then match the functions with their graphs shown below using your limit results Some will require you to compute left and right hand limits 1 a lim 1 57 5 b lim T H x 5 x 2x 35 x 5 x 4x 5 c lim d lim x 5 x 3x 10 x 5 Week 4

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

Math 124 4 Find if any the horizontal asymptotes of the following functions and use that information to match them with their graphs on the next page Each question should have two limit computations with and r o x 1 4 a f x r 3x 7x 10 Worksheet x 3 b f x 2 8x 26 c f x x 4x 9 x 4

f a h f a h h 0 3 Use the definition of the derivative f a lim to compute f 3 for the following functions Then match the functions with their graphs shown below using your limit results a f x x 3 2 b f x x 3 2 c f x x 3 2

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

B 15 Let m and b be real numbers and let the function fbe defined by f x 1 3bx 2x for x 1 mx b for x 1 Iffis both continuous and differentiable at x 1 then A m 1 b 1 B m 1 b 1 D m 1 b 1 2 C m 1 b 1 E none of these

2 Let f be the function given by f x x 3x x 2 A The tangent to the graph of fat the point P 2 8 intersects the graph off again at the point Q Find the coordinates of the point Q B Find the coordinates of point R the point where f x 0 6 Show that the segment QR divides the region between the graph of fand its 16 tangent at P into two regions whose areas are in the ration of 11 2 8 P2

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

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 4t 6 m sec over the time interval 0 5 Enter your answer as an exact fraction if necessary and do not include units in your answer

For the region bounded by y 3x the x axis and x 5 determine which of the following is greater the volume of the solid generated when the region is revolved about the x axis or about the y a When the region is revolved about the x axis the volume is Type an exact answer using as needed Points 0 of 1 I

Find the volume of the solid generated when R shaded region is revolved about the given line y 18 x y 18 x 324 about y 18 The volume of the solid obtained by revolving th

Find the volume of the solid generated when R shaded region is revolved about the given line x 4 2 secy x 4 y T 3 and y 0 about x 4

Find the volume of the solid generated when the region bounded by y 4x and y 12 x is revolved about the x axis Set up the integral that gives the volume of the solid 10 dx CID YA

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

Use part I of the Fundamental Theorem of Calculus to find the derivative of 8x u 5 g x du u 3

14 A sign is hanging from a storefront The two ropes holding it are fixed at angles of 25 and 35 with the horizontal If the sign has a weight of 1150 N what is the tension in each rope A 3 T 2

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

Find the volume of the solid generated when the region bounded by y 2x and y 4 x is revolved about the x axis Set up the integral that gives the volume of the solid dx

Find the volume of the solid generated by revolving the region R bounded by y ex y 0 x 0 and x In 3 about the x axis Set up the integral that gives the volume of the solid In 3 z e 2x dx Type exact answers The volume of the resulting solid is 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

ind the volume of the solid generated when the region bounded by y 3x and y 6 x is revolved about the x axis Set up the integral that gives the volume of the solid To 0 dx y 6 x y 3

Find the volume of the solid generated when the region bounded by y 3x and y 12 x is revolved about the x axis Set up the integral that gives the volume of the solid 9

Question Find the total distance traveled by a particle according to the velocity function v t 4t 4 m sec over the time interval 0 4 Enter your answer as an exact fraction if necessary and do not include units in your answer Provide your answer below

Find the volume of the solid generated by revolving the region bounded by y 2x y 0 and x 3 about the x axis Set up the integral that gives the volume of the solid 10 dx 0 1 epowers CIT

Question 2 If 8 f x dx 3 and S f x dx 2 what is the value of P f x da 8 5

If fog x h x such that x 4x 8 which of the following could accurately represent f and g Of x 2x 4 and g x 2x 4 Of x 4x 8 and g x x Of x x and g x 4x 8 Of x 2x 4 and g x 2

Question Determine the value of S Provide your answer below fox f x dx given that 6 f x dx 9 and f x dx 3

Guess the value of the limit lim x18 by evaluating the function f x for x 0 1 2 3 4 5 6 7 8 9 10 20 50 and 100 Use a graph off to support y

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

Tutorial Exercise A ball is thrown into the air with a velocity of 46 ft s Its height in feet after t seconds is given by s t 46t 16t2 Find the velocity of the ball at time t 2 seconds Step 1 Let h s t 46t 16t2 give the height of the ball at time t Then the ball s velocity at time t a can be found by s t s a t a v a lim t a We are asked to find the velocity at t 2 therefore we use a v 2 lim s t s 2 t 2 t 2 lim t 2 lim t 2 46t 16t 46 16 2 46t 16t2 t 2 t 2 and have

Question A particle travels according to the velocity function v t 4t 10 m sec What is the total distance traveled by the particle over the time interval 2 12 Enter your answer as an exact fraction if necessary and do not include units in your answer

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

If a rock is thrown upward on the planet Mars with a velocity of 11 m s Its height above the ground in meters after t seconds is given by H 11t 1 86t a Find the velocity in m s of the rock after 1 second 7 28 m s b Find the velocity in m s of the rock when t a 3 72 X m s c When in seconds will the rock hit the surface Round your answer to one decimal place t 5 91 S d With what velocity in m s will the rock hit the surface 4 08 X m s

Find the horizontal and vertical asymptotes of the curve You may want to use a graphing calculator or computer to check you DNE X 11 y 4x x 1 x x 6