Calculus Questions

Find the area of the sector shown in the figure 5x 6 OU The area is Simplify your answer Type an exact answer in terms of Use integers or fractions for any numbers in the expression

90 horsepower outboard motor at full throttle will rotate its propeller at exactly 4500 revolutions per min Find the ngular speed of the propeller in radians per second 500 rev per min radians per sec Round to the nearest tenth as needed

Directions The displacement in meters of a particle moving in a straight line is given by the following equation where t is measured in seconds 1 Find the velocity of the particle after 3 seconds 2 Find the speed of the particle after 3 seconds Note You do not need to enter the units m s m s s t 52 t2

A population of cattle is increasing at a rate of 600 80 t per year where t is measured in years By how much does the population increase between the 6 th and the 9 th years Total Increase 240

Use the formula to find the value of the missing variable 0 radians t 9 sec radians per sec Simplify your answer Use integers or fractions for any numbers in the expression Type an exact answer as needed

Use the formula s rot to find the value of the missing variable T 4 radians per sec r 12 cm t 16 sec S cm Type an exact answer using as needed

Evaluate the indefinite integral S xl x 4 5x dx

The tangent line to the graph of y x at the point x y has slope 3x Using this result find the slope of the curve at the point 3 27 The slope of the curve at the point 3 27 is

1 1 5 sec

Use the Evaluation Theorem to decide if the definite integral exists and either evaluate the integral or enter DNE if it does not exist 57 Lot secx tan x dx

Evaluate the indefinite integral du 6u4 I

Evaluate the indefinite integral 7 dx

Let L be the line through points P and Q in the figure a Suppose P 4 2 and Q 8 5 Find the slope of L and the length of d b As the point Q moves towards P along the curve does the slope of L increase or decrease a The slope of Lis d

Convert the radian measure to degrees 2 radian s

Evaluate the definite integral 4 1 6 5x dx 0

5 15r Gr

Find the point on the graph of y x where the curve has slope m 5 The point on y x at which the tangent line has slope 5 is Type an ordered pair Use integers or fractions for any numbers in the expression

Here is a graph of f x x2 and the line tangent to f x at a f a The equation of the tangent line is y 2x 1 Find a f a and the slope of the parabola f x at a f a X

Find the point s on the graph to the right where the slope is equal to 6 11 The point s on the given graph at which the slope is equal to F is are Type an ordered pair Type an exact answer for each coordinate using radicals as needed Use s y x3 y x y X Slope is 3x2

Convert the following radian measure to degree measure 4x 4x 3 11

Convert the following degree measure to radian measure 87 87 radians Type an integer or a decimal Round to the nearest hundredth

Convert the following degree measure to radians 168 30 168 30 radians Round to the nearest thousandth as nee

Convert the following degree measure to radian measure 65 15 65 15 radians Type an integer or a decimal Round to the nearest thousandth as needed

Convert the following degree measure to radians 270 270 radians Simplify your answer Type an exact answer using as needed

Find the complementary and supplementary angles to 12 T 12 12 Simplify your answer Type a fraction with in the numerator The complementary angle to 5x The supplementary angle to is 5x 12 Simplify your answer Type a fraction with in the numerator is

Sketch the angle in standard position and draw an arrow representing the correct amount of rotation Find the measure of two other angles one positive and one negative that are coterminal with the given angle and closer to the given angle than any other coterminal angle Give the quadrant of the angle if applicable A 62 Choose the correct graph below where the curve on each graph traces the angle beginning at the positive x axis and ending at the ray O A Ay 0 62 5 OB Ay Q O C

Convert the following angle to degrees minutes and seconds form a 83 7022 The answer is 0 Simplify your answers Round to the nearest second as needed

Convert the following angle to decimal degree notation 31 9 31 9 Round to two decimal places as needed

Perform the calculation 69 12 52 42 69 12 52 42 0 Simplify your answers Type an integer or a fra

Find the slope of the line tangent to the curve y x at the point The slope is Tyne an integer or a simplified fraction 9 81 4 16 and then find the corresponding equation of the tangent line

Write the equation of the tangent line to the graph of y x at the point x 0 9 Tyne an equation

Estimate the slope of the following curve at the designated point 10 8 7 6 5 4 1 0 1 2 3 A 5 6 7 8 9 X Q COLED What is your estimate of the slop slope Round to the neare

1 A tank holds 1000 gallons of water which drains from the bottom of the tank in half an hour The values in the table show the volume V of water remaining in the tank in gallons after t minutes t min V gal 5 694 444 250 111 10 15 20 25 28 30 0 a If P is the point 15 250 on the graph of V find the slopes of the secant lines PQ when Q is the point on the graph with t 5 10 20 25 and 30 b Estimate the slope of the tangent line at P by averaging the slopes of two secant lines c Use a graph of V to estimate the slope of the tangent line at P This slope represents the rate at which the water is flowing from the tank after 15 minutes

Directions If a rock is thrown upward with a velocity of 30 m s it has a height in meters after t seconds given by h t 30t 4 9t 1 Find the velocity of the rock after 6 seconds 2 When will the rock hit the surface Note You do not need to enter the units seconds m s

Directions Consider the following function where A and B are constants Find these constants A and Part 1 Difference Quotient If x 1 then the difference quotient can be simplified into the form Ax B That is f x f 1 x 1 and B IL Note It s possible for one or more of these constants to be 0 f x 5x 6 f 1 n Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 1 Ax B f 1 lim f x f 1 x 1

Directions Consider the following function f x at a 3 Part 1 Difference Quotient A If h0 then the difference quotient can be simplified into the form f 3 h f 3 h where A B C D and E are all non negative constants Find these constants B C 0 0 D Note It s possible for one or more of these constants to be 0 E 5 x A Bh C Dh Eh A Bh C Dh Eh f 3 lim h 0 Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 3 That is note the negative in front f 3 h f 3 h

Directions Consider the following function f x at a 8 Part 1 Difference Quotient If h 0 then the difference quotient can be simplified into the form f 8 h f 8 8 h where A and B are constants Find these constants A and B Note It s possible for one or more of these constants to be 0 f x x 7 f 9 1 VA h B Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 8 f 8 lim h 0 That is 1 A h B f 8 h f 8 h

Directions Use the graph of function f x given below to estimate the following values of its derivative J x Note that a portion of the tangent line has been drawn for you at x 2 1 Estimate the value of f 4 2 Estimate the value of f 2 3 Estimate the value of f 1 ello 5 4 f x 2 3 4 5 H

Find an equation of the tangent line to the given curve at the point 1 0 Tangent Line y y x 1 x x 1

Calculate the derivative of the function 71 H u u V u v 1

Use the graph of f x given below to evaluate the following limits at infinity 1 lim f x 2 x 2 lim f x If necessary use inf and inf to denote o and co and DNE if the limit does not exist

Part 1 Difference Quotient If h 0 then the difference quotient can be simplified into the form That is f x f x h f x h Note It s possible for one or more of these coefficients to be 0 1 x 4 1 x 4 A Bh 1 x 4 A Bh where the coefficients A and B can be functions of a or just constant terms Find these coefficients A B Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f x f x lim h 0 f x h f x h

Directions Consider the following function f x f x 3x 10 Part 1 Difference Quotient If h 0 then the difference quotient can be simplified into the form A Bh Ch That is f x h f x h A Bh Ch where the coefficients A B and C can be functions of a or just constant terms Find these coefficients A 0 B 3 C 0 Note It s possible for one or more of these coefficients to be 0 Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f x f x lim h 0 f x h f x h

Determine the area of the shaded region in the figure What is the area of the region in terms of x 6 OA x 15 dx 15 10 OC S x 15 15 10 S x 15 6 dx dx x y 15 x 2y Q o G COLLE OB Points 0 c 10 S 22 x 15 dx 15 6 10 D 2 f x 15 dx f x 15 15 6

Find the intervals on which f is increasing and the intervals on which it is decreasing f x cos x on Select the correct choice and if necessary fill in the answer box es within your choice A The function is increasing on the open interval s and decreasing on the open interval s Simplify your answers Type your answers in interval notation Type exact answers using as needed Use a comma to separate answers as need OB The function is increasing on the open interval s The function is never decreasing Simplify your answer Type your answer in interval notation Type exact answers using as needed Use a comma to separate answers as needed OC The function is decreasing on the open interval s The function is never increasing Simplify your answer Type your answer in interval notation Type exact answers using as needed Use a comma to separate answers as needed OD The function is never increasing or decreasing

Determine the area of the shaded region bounded by y x 4x and y x 2x 30 20 10

If a 1 2 6 6 and b 3 2 7 find a b Your Answer Answer Which statement best describes the result of the cross product a It gives the magnitude of the projection of one vector onto another b It defines a vector perpendicular to both given vectors c The cross product gives a vector parallel to both given vectors d It is the division of one vector by another If A 3 2 9 and B 4 7 2 are two points in 3 space then find the vector AB a AB 7 5 7 b AB 1 9 11 c AB 12 14 18 d AB 7 5 71 The dot product represents a The magnitude of the projection of one vector onto another b A vector perpendicular to both given vectors Oc A vector parallel to both given vectors d The division of one vector by another What is 0 0 1 x 0 1 0 a 0 1 0 b 0 0 1 c 1 0 0 d 1 0 0

region in two ways a Using integration with respect to x b Using geometry OC 2 6 x dx 0 OD 6 S 0 6 x x dx b Using geometry which would be the appropriate set up for the area OA A 2 6 3 1 OB A 1 2 6 6 6 3 c A OD A 2 6 6 The area of the shaded region is

or wages less than the maximum taxable wage base Social Security contributions including those for Medicare by employees are 7 65 of the employee s wages T a Find an equation that expresses the relationship between the wages earned x and the Social Security taxes paid y by an employee who earns less than the maximum taxable wage base y b For each additional dollar that an employee earns by how much is his or her Social Security contribution increased Assume that the employee s wages remain less than the maximum taxable wage bas c What Social Security contributions will an employee who earns 55 000 which is less than the maximum taxable wage base be required to make

Find the area of the region in two ways a Using integration with respect to x b Using geometry a Using integration with respect to x what would be the appropriate integral to find the area 10 OA 10 4 10 x x 4 dx 3 OB 2 10 x dx 2 10 x 0 OC 2 3 2 f x 4 dx 0 3 OD 10 x x x 4 dx 10 8 6 4 2 Ay 0