Calculus Questions

Compute the following derivatives a b d da d da r arcsin r sin r arccos r

Let f x cos arcsin a a First find the derivative by using Chain Rule f x b Next we will find the derivative in a different way Rewrite f x without trigonometric functions Hint Set up a right triangle with a side x When you have finished this step f x can be written as f x Then find the derivative of this equivalent form of f

Find the equation of the line tangent to the graph of f x at the indicated x value f x 3 sin 5x x 0 Tangent line y

18 QUESTION 18 Solve the problem A fossilized leaf contains 35 of its normal amount of carbon 14 How old is the fossil to the nearest year Use 5600 years as the half life of carbon 14 28 672 3474 8466 33 664 19 20

If x y 90 and 9 find when y 3 and x 9

dr Let A be the area of a circle with radius r If dt dA dt 2 find when r 2

QUESTION 15 Solve the equation log3 x log3 x 24 4 No real solutions 27 O 53 O 3 27

The graph of a logarithmic function is shown Select the function which matches the graph Oy log 1 x Oy log x 1 Oy 1 log x

QUESTION 21 The function f is one to one Find its inverse f x 6x 7 x 7 f x x 7 f1 x 2 6 7 f1 x 2 7 O O f x x 7

QUESTION 20 Solve the problem Conservationists tagged 60 black nosed rabbits in a national forest in 1990 In 1993 they tagged 120 black nosed rabbits in the same range If the rabbit population follows the exponential law how many rabbits will be in the range 10 years from 1990 1210 605 74 148 QUESTION 21

QUESTION 22 The function f is one to one Find its inverse f x x 6 x 0 O f x x 6 x 6 Of x x 6 x 0 Of x x 6 x 6 f x x 6 x 0

QUESTION 24 Use the given zero to find the remaining zeros of the function f x x 3x 5x 39 zero 3 3 2i 3 2i 1 2i 1 2i 1 2 131 1 2 13i 6 4i 6 4i QUESTION 25

Form a polynomial whose zeros and degree are given Zeros 0 5 4 degree 3 f x x x x 20 for a 1 Of x x3 x2 20x for a 1 Of x x3 x2 20x for a 1 f x x x x 20 for a 1

Find the power function that the graph of f resembles for large values of Ix f x x x 9 x y x7 y x7 y x3 Oy x QUESTION

Find the vertical asymptotes of the rational function f x x2 16 x2 5x 4 x 1 x 4 X 1 X 4 x 1 x 4 x 1

Given f x 4x 5 g x 5x 1 20x 9 20x 24 20x 1 20x 4 Find f g x

Form a polynomial f x with real coefficients having the given degree and zeros Degree 3 zeros 3 and 3 2i Of x x3 x2 5x 39 f x x 3x 5x 39 f x x3 x2 11x 39 Of x x3 3x 5x 52

Find the x and y intercepts of f f x x 1 x 4 x 1 Ox intercepts Ox intercepts Ox intercepts Ox intercepts 1 1 4 y intercept 4 1 1 4 y intercept 4 1 1 4 y intercept 4 1 1 4 y intercept 4

Determine the maximum number of turning points of f f x x 2 x 5 3 0 000

QUESTION 1 Decide whether or not the functions are inverses of each other f x 7x 4 g x Yes No 7

Find the domain of the rational function 2x 4 3x2 6x 45 f x x x 3 x 3 x 5 x x 3 x 5 x x 3 x 5 all real numbers

Find the domain of the function f x In 4x x 145x 0 44x44 0 x 4 Ox 4

Evaluate the expression using the values given in the table g f 1 X 1 71012 5 2 1 3 14 f x 2 10 1 14 g x 1 87 10 800 1 3 2 13 10 2 7 8 X

Find the domain of the composite function f g 0000 f x g f x g x 63 X 9 x x 0 x 9 x 7 x x 0 x 7 x x is any real number x x 0 x 9

Which of the following would not be a representation of the differentiable function f where f 5 3 6 points The slope of y f x at x 5 is 3 The instantaneous rate of change at x 5 of f x is 3 lim h 0 f 3 h f 3 h 5 The slope of a line tangent to the graph of fat x 5 is 3 Which of the following would not be a representation of the differentiable function f where f 5 3 6 points O The slope of y fx at x 5 is 30 The instantaneous rate of change at x 5 of f x is 3 Of 3 hf 3 lim h 0 h 5 The slope of a line tangent to the graph of f at x 5 is 3 Previous Question Next Question

2 A Given the following information about the first and second derivatives of a function of that is continuous on indicate all possible critical points the locations of the local max min and the inflection points and then sketch a possible graph of f 0f 0 f 0f 0 f 0f 0 f 0f 0 f 0 f 0 f 0f c I

Determine r x if r x ln 3 X COS X 7x 6

x 2 103 Find the derivative of f x 5x 1

05 Find the equation of the tangent line to the graph of f x when z 0

Here is a graph of the function g V 5 4 3 60 4 w 24 1 H 24 3 4 y 4 fo 5 X Use the graph to find the following If there is more than one answer separate them with commas a All values at which g has a local maximum b All local maximum values of g

be an absolute minimum value for f x on the interval 1 4 34 Find the absolute maximum of f x x4 8r on the interval 0 7 11 B

101 Find f x of f x g z dy da 102 Find if y 5log3 1

concave down and find any r values of inflection points 32 Find the x values of the inflection points of f x x 18x

A parcel delivery service has contracted you to design a closed box with a square base that has a volume of 11 000 cubic inches Complete parts a d M A X X a

lath 126 3 12 pts a What is the average rate of change of the function f x 1 the equation of the secant line through 3 f 3 and 1 2 f 1 2 reduced fractions c Sketch a graph of f and the secant line from 3 to 1 2 b What i Write your answers in terms

Find all real zeros of the function f x 6x 19x 45x 100 Given 4 is a factor of f x Enter the real zeros separated by commas Give as exact values Question Help Mec

The half life of cesium 137 is 30 years Suppose we have a 180 mg sample a Find the mass in mg that remains after t years y t b How much of the sample in mg remains after 80 years Round your answer to two decimal places X mg c After how many years will only 1 mg remain Round your answer to one decimal place yr t mg Enter a number AWA

A storage tank is being designed to accomodate 1 000 m of contaminated water for a mining operation The storage tank must be cylindrical with a radius R and height H and will be composed of three steel sections Flat disks as the floor and ceiling and the curved portion comprising the side Find the dimensions R and H that minimize the steel used

60 50 40 30 20 10 10 Type answer below 20 61718 1 30 40 9 50 10 1 11 12 113 114 115 1 16 Maximum output for TC of 400 000 60 a State the least cost combination of inputs for this firm TPP 4 500 TC 400 000 Labour L www economichep org TPP 4 000 TPP 3 500 2 marks b State the production level that will be produced using the least cost method 1 mark 17 181 19

a The domain of f x 2 4 b The range of f x 1 3 2 1 The graph of y f x given above starts at x 2 and ends at x 4 Use the graph of y f x to identify a 2 473 c f 0 I d For what values of x does f x 1 Doesn t wom e On what interval s is f x 0 2 4 f How many 3 3 2 2 4 1 b 1 33 d A e nes does the line y intersect the graph of f x f Q4 and

2 1 a The domain of f x 2 4 b The range of f x 1 3 The graph of y f x given above starts at x 2 and ends at x 4 Use the graph of y f x to identify 2 47 c f 0 1 2 d For what values of x does f x 1 Doesn t wom 2 2 a b 1 3 C d M

1a b C Evaluate f 3 for f x 3x 7x 4 6 Find the third derivative of f x 3x 7x 4 Find the point on the parabola y x 3x 4 where the slope of the tangent is 4 If V represent the volume of an object what does represent d DATOSESI ORCHAEOL C4 3

Find the derivative of the following a f x x 3x 2 x 5 b f x C d e f f x 132 f x 10x f x 2 36 y x f x 3 2x 5x 4 2x 3 4 3 7x 4x 3x 1 2x 1 2 9 f x x 3 1 2x g www 2 11x 12x 16

se implicit differentiation to find dy dx 15 x3 3x2y y3 8 x2 3xy x y2 A B x2 2xy x2 y2 x2 2xy C x2 y2 D x2 32 x2 y

9 O O O D On what interval s is f increasing and on what interval s is f decreasing Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function is increasing on The function is never decreasing Simplify your answer Type your answer in interval notation Type an exact answer using radicals as needed Use a comma to separate answers as needed OB The function is decreasing on The function is never increasing Simplify your answer Type your answer in interval notation Type an exact answer using radicals as needed Use a comma to separate answers as needed OC The function is increasing on and decreasing on Simplify your answers Type your answers in interval notation Type exact answers using radicals as needed Use a comma to separate answers as needed OD The function is never increasing or decreasing Determine the coordinates of the relative extrema Select the correct choice below and if necessary fill in the answer box to complete your choice O Sketch the graph of the following function Indicate where the function is increasing or decreasing where any relative extrema occur where asymptotes occur where the graph is concave up of conca O f x 2 x 7 OA The coordinates of the relative extrema are Type an ordered pair Type an exact answer using radicals as needed Use a comma to separate answers as needed OB There are no relative extrema Determine the vertical asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has one vertical asymptote Type an equation OB The function has two vertical asymptotes The leftmost asymptote is Type equations OC The function has no vertical asymptotes Determine the horizontal asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has two horizontal asymptotes The top asymptote is and the bottom asymptote is Type equations OB The function has one horizontal asymptote Type an equation OC The function has no horizontal asymptotes Determine the slant asymptote s Select the correct choice below and if necessary fill in the answer box to complete your choice OA The stant asymptote s is are Type an equation Use a comma to separate answers as needed OB The function has no slant asymptotes and the rightmost asymptote is

According to a study a vehicle s fuel economy in miles per gallon mpg decreases rapidly for speeds over 60 mph a Estimate the speed at which the absolute maximum gasoline mileage is obtained b Estimate the speed at which the absolute minimum gasoline mileage is obtained c What is the mileage obtained at 70 mph C Fuel economy mpg 36 30 24 18 12 6 55 65 75 85 95

Sketch the graph of the following function Indicate where the function is increasing or decreasing where any relative extrema occur where asymptotes occur where the graph is concave up or concave down where any points of inflection occur and where any intercepts occur f x 2 x 7 On what interval s is f concave up and on what interval s is f concave down Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function is concave up on and concave down on Simplify your answers Type your OB The function is concave up on Simplify your answer Type your OC The function is concave down on and is never concave up Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed Determine the coordinates of the point s of inflection Select the correct choice below and if necessary fill in the answer box to complete your choice answers in interval notation Use a comma to separate answers as needed and is never concave down answer in interval notation Use a comma to separate answers as needed OA The coordinates of the point s of inflection are Simplify your answer Type an ordered pair Use a comma to separate answers as needed OB There are no points of inflection Determine the x intercept s Select the correct choice below and if necessary fill in the answer box to complete your choice OA The x intercept s is are Type an ordered pair Type an exact answer using radicals as needed Use a comma to separate answers as needed OB There are no x intercepts Determine the y intercept s Select the correct choice below and if necessary fill in the answer box to complete your choice OA The y intercept is Simplify your answer Type an ordered pair Use integers or fractions for any numbers in the expression OB There is no y intercept Sketch the graph of the function Choose the correct graph below OA AY Q OB Ay GILD Q OC Q O D Q

For the graph shown identify a the point s of inflection and b the intervals where the function is concave up or concave down a The point s of inflection is are Type an ordered pair Use a comma to separate answers as needed b Identify the intervals where the function is concave up or concave down Z 10 O N The function is concave up on and concave down on Simplify your answers Type your answers in interval notation Use a comma to separate answers as needed

For the following function a give the coordinates of any critical points and classify each point as a relative maximum a relative minimum or neither b identify intervals where the function is increasing or decreasing c give the coordinates of any points of inflection d identify intervals wher up or concave down and e sketch the graph h x 2x 24x a What are the coordinates of the relative extrema Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The relative minimum point s is are and the relative maximum point s is are Simplify your answers Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed OB The relative maximum point s is are and there are no relative minimum point s Simplify your answer Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed and there are no relative maximum point s OC The relative minimum point s is are Simplify your answer Use integers or fractions for any numbers in the expression Type an ordered pair Use a comma to separate answers as needed OD There are no relative minimum points and there are no relative maximum points b On what interval s is h increasing or decreasing Select the correct choice below and if necessary fill in the answer box es to complete your choice A The function is increasing on The function is decreasing on Simplify your answers Type your answers in interval notation Use a comma to separate answers as needed OB The function is decreasing on The function is never increasing Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed OC The function is increasing on The function is never decreasing Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed OD The function is never increasing or decreasing c What are the coordinates of the inflection point s Select the correct choice below and if necessary fill in the answer box to complete your choice OA The coordinates of the inflection point s are Simplify your answer Type an ordered pair Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB There are no inflection points d On what interval s is h concave up or concave down Select the correct choice below and fill in the answer box es to complete your choice Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed OA The function is concave up on and concave down on OB The function is concave up on OC The function is concave down on GIIB and is never concave down and is never concave up

Determine the horizontal asymptote of the function If none exists state that fact 4x 5x 8x X Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has one horizontal asymptote Type an equation OB The function has two horizontal asymptotes The top asymptote is Type equations OC The function has no horizontal asymptotes and the bottom asymptote is