Calculus Questions

6 Apply the Intermediate Value Theorem to determine on which of these intervals the graph of f x 2x 7x 1 must intersect the line y 4 A 8 6 B 4 1 C 0 2 D 6 9 E Can t guarantee that it will intersect

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 1 01 X Hint If the throw starts at t 1 and lasts 01 sec what will the ending time t2 be 005 sec 0 005 X 002 sec 46 4 001 sec 0 001 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

Population Growth The graph of P t below shows the population in thousands of people of a city t years after 2008 Note that 8 years after 2008 the population is approximately 3 7 thousand people And 16 years after 2008 the population is approximately 6 85 thousand people Approximate the average rates of change over each of the following time intervals Round to two decimal places 0 years to 8 years 7125 X thousand people per year 8 years to 16 years 1 3125 X thousand people per year 0 years to 16 years thousand people per year set 2 wido 2 13 12 7 8 9 10 11 12 13 14 15 16 Years since 2008 Complete the statement to explain the meaning the rate of change in this situation The city s population is increasing at an inconsistent What do the values above tell you about the rate of change of V t The rate of change for D t is Non linear rate

Use the graph of f to evaluate the following 6 5 1 4 3 Next question 2 1 14 Get a similar question 8 1 2 3 4 5 6 The average rate of change of f from 0 to 4 Zeman 3 4 You can re X

Given f x 1 find the average rate of change of f x on the interval 5 5 h Your answer will x 8 be an expression involving h

10 81 4 2 6 4 1 2 2 4 2 A 4 6 8 10 12 6 Clear All Draw Dot Line Segment a Mark the two points on the graph that are needed to calculate Average Rate of Change over the interval 2 3 b Connect the points with a line segment c The Average Rate of Change over 2 3 is

Use the graph of f to evaluate the following 1 6 5 4 3 2 1 1 f x 2 3 4 5 X of The average rate of change of f from 0 to 4

Given f x x 7 find an expression that gives the slope of the line from 2 2 to 2 h f 2 h Your answer will be an expression involving h

Let x 2 2 A Sketch the graph of f and when you re done place a 1 in the box values B Find the value of a where f is discontinuous If there is no value enter NONE x 0 x 0 values C Find the value of a where f is not differentiable If there is no value enter NONE

A spherical balloon is being inflated Find the rate of increase with respect to its radius in inches r of the surface area in square inches S 4T when A r 2 inches Rate of increase in square inches per second B 4 inches Rate of increase in square inches per second C r 5 inches Rate of increase in square inches per second

At what point does the line normal to the graph of y 3x 3 4x at 1 4 intersect the parabola a second time 0 Hint The normal line is perpendicular to the tangent line If two lines are perpendicular their slopes are negative reciprocals i e if the slope of the first line is m then the slope of the second line is 1 m

2 Write the equation of a line that is perpendicular to y 8 x 1 and passes through 4 1

Find equations of the tangent line and of the normal line to the curve y 5 2x 2 at the point 1 49 Write the equations of the lines in the form y mx b Tangent line y Normal line y

For what value s of a is the tangent line to the graph of parallel to the line y 2x 0 4 f x 4x 6x 142x 48 If there is more than one value of a list them as a comma separated list

Let f x 8 Then the expression f x h f x h where A is a constant and A I Using your answer from above we have f x h f x f x lim h Finally find each of the following 1 m 2 3 can be written in the form A x x h

Let f x x 8 Evaluate the following limits T lim Z 8 lim f x f 8 x 8 CAMENTE f x 8 x 8 Thus the function 2 P ENCO

Find the equation of the tangent line to the curve y 6 sin x at the point 6 3 z The equation of this tangent line can be written in the form y mx b where and b

Find the points on the graph of f x 2x 2x 3 List the x values of these points I value s Separate answers by commas if ther 6x 6x 90x 24 where the tangent is horizontal

Find the first and second derivatives of the function f x x FORRE f z 2sin2 4cost

graph by using the vertical line test If it is use the graph to find a The domain and range b The intercepts if any c Any symmetry with respect to the x axis the y axis or the origin IYU A The domain is 00 The range is Type your answers in interval notation C 18 X 10 12 a If the graph is that of a function what are its domain and range Select the correct choice below and h if necessary fill in the answer box to complete your choice

65 3 2 1 0 N 3 6 m x on 2 9 3 m x 4 5 6 ns sild zizil sode GLOS VOOS moral 8 9 10 OTH

a f 0 e f x Wing for the function f x 4x 3x 4 b f 5 f f x 3 a f 0 4 Simplify your answer b f 5 111 Simplify your answer c f 5 81 Simplify your answer c f 5 g f 5x 2 d f x 4x 3x 4 Simplify your answer e f x 4x 3x 4 Simplify your answer f f x 3 4x 27x 41 Simplify your answer 2 g f 5x 100x 15x 4 Simplify your answer h f x h Simplify your answer d f x h f x h K

Find the following for the function f x 4x 3x 4 b f 5 a f 0 e f x c f 5 g f 5x f f x 3 a f 0 4 Simplify your answer b f 5 111 Simplify your answer c f 5 81 Simplify your answer 2 d f x 4x 3x 4 Simplify your answer 2 e f x 4x 3x 4 Simplify your answer f f x 3 4x 27x 41 Simplify your answer 2 g f 5x 100x 18 4 Simplify your answer

8 Let n x x 4 The average rate of change of n x over the interval c 5 is equal to 3 where c is a constant Find the value of c 70 navig ont 25 no

Find the following for the function f x 4x 3x 4 b f 5 c f 5 a f 0 e f x f f x 3 g f 5x a f 0 4 Simplify your answer b f 5 111 Simplify your answer c f 5 81 Simplify your answer 2 d f x 4x 3x 4 Simplify your answer 2 e f x 4x 3x 4 Simplify your answer f f x 3 Simplify your answer d f x h f x

Find the following for the function f x a f 0 e f x X b f 2 f f x 2 2 c f 2 5 Simplify your answer Type an integer or a simplified fraction x 1 X f f x 2 c f 2 g f 4x d f x 2 x 1 Simplify your answer Use integers or fractions for any numbers in the expression d f x h f x h e f x 2 X 1 Simplify your answer Use integers or fractions for any numbers in the expression x 2 x 4 17 Simplify your answer Ilse integers or fractions for any numbers in the expression

3V 33 The radius r in inches of a basketball is related to the volume V by r V 4 into the basketball so the volume after t seconds is given by V t 24 Find the composite function V 1 a 25 Find the time when the radius reaches 9 inches Air is pumped

a f 0 e f x b f 3 f f x 3 c f 3 g f 7x a f 0 14 Simplify your answer Type an integer or a fraction b f 3 17 Simplify your answer Type an integer or a fraction c f 3 Simplify your answer Type an integer or a fraction d f x h f x h

Find the following for the function f x x 14 b f 3 a f 0 e f x f f x 3 c f 3 g f 7x a f 0 14 Simplify your answer Type an integer or a fraction b f 3 Simplify your answer Type an integer or a fraction d f x h f x h

Sylvanite is a mineral that contains 28 0 gold by mass How much sylvanite would you need to dig Express your answer with the appropriate units View Available Hint s Value Submit Part C 1220 Units If an athlete weighs 141 lb and has 15 0 body fat by weight how many pounds of fat does the athlete s body contain Express your answer with the appropriate units View Available Hint s

1 1 y y at Final Position m 82 44 102 51 163 46 286 88 461 45 722 53 let y letx T Yotat 17 92x 23 962 Plot a graph for this data and determine the acceleration Time sec 6 7 9 12 15 20 Y 2 o x yo Eq n in y mx b form

A 11 630 g milk chocolate bar is found to contain 7 815 g of sugar Part A How many milligrams of sugar does the milk chocolate bar contain Express your answer in milligrams to four significant figures View Available Hint s 7815 mg Bull Comit Previous Answers Correct The unit analysis method is used in everyday activities This method is very useful in quantifying the terms and making conversions used frequently in the analysis of different substances Part B What will be the amount of sugar in milligrams if the size of the milk chocolate bar is reduced from 11 630 g to 5 000 g Express your answer in milligrams to four significant figures View Available Hint s

A golf ball is hit with a driver and neglecting almost all physical forces except gravity follows the path y 300 x I 600 2 The ball is on the ground at x 0 and at x 300 We will assume that the ball did not roll when it hit the ground We would normally say that the drive was 300 yards long which is the distance along the ground that the ball traveled The actual distance that the ball travels is given by the arc length of the parabola The arclength is given by the integral 300 1 y dx 1 x 300 1 2 dx Problems 1 Evaluate the integral above using an electronic device How far did the ball travel 2 The graph of a function f x is given below It consists of three line segments for 0 x 8 Each mark along the x axis and y axis is 1 unit a Calculate the total length of f x for 0 x 8 b What is f x for 0 x 3 c What is f x for 3 x 4 d What is f x for 4 x 8 f x e The graph of y 1 f x is given below Calculate the y coordinates of the three line segments using your answers from parts b d f Find the areas of the three rectangles in the second figure below using your answers from part e 1 f x

4 In 3 4 sentences describe the main similarities and differences between prevalence and incidence Consider both what information goes into the formula but also how or why that information is important and could be used

Given that a 11 630 g milk chocolate bar contains 7 815 g of sugar calculate the percentage of sugar present in 11 630 g of milk chocolate bar keeping in mind that the answer should have four significant figures two decimal places Express your answer numerically to four significant figures View Available Hint a

nd the intervals on which the function is continuous 17 y 2 X 7 2x A continuous everywhere C discontinuous only when x 9 B discontinuous only when x D discontinuous only when x

ind all vertical asymptotes of the given function 9 16 g x 2 25 A none C x 5 x 5 x 9 B x 5 x 5 D x 5 x

14 lim h 0 x h 3 x3 h A 3x2 3xh h B Does not exist C 3x2

Find the limit if it exists x2 100 X 10 13 lim X 10 A 1 B 20 C 10 D Does not exist

13 Using the washer method determine the volume of the solid obtained by rotating the region bounded by y 2 x 1 and y x 1 about the line x 1 1 0 1 2 3 7 6 5 4 3 1 2 3 4 5 X

11 Use the slicing method to find the volume of a pyramid with a square base sides of 3 and height of 5 do not use the formula memorized from geometry 3 y

17 1 point The two quadratic functions have the same vertex f x 3 x 2 3 f x x 4x 1 True False

Determine the limit by sketching an appropriate graph 9 x2 0 x 3 3 x 5 X 5 12 lim f x where f x X 5 A 3 3 5 B 0 C Does not exist D

7 y 2x 2 8 A 37 100 B 2 1 3 C D

Let Horizontal asymptotes y f x Find the equations of the horizontal asymptotes and the vertical asymptotes of f x If there are no asymptotes of a given type enter NONE If there is more than one asymptote of a given type give a comma seperated list i e 1 2 Vertical Asymptotes x x 3x 4 3x 11x 4

Evaluate the following limits 2 lim I 3 I 3 1 1 lim 17 x x 7 2 lim 2 3 S 3 1 lim x 0 x x 7 DNE X DNE

Let f x arsitate 3x57x4 7z4 3 Then

Let x 5 x x 4 x 6 Then TOETUST 3

If f t 7 t f t 9 vt find f t

If f u 2u 5u find f u