Application of derivatives Questions and Answers

Examples 1 Find the slope of the tangent to the ellipse x 4y 25 at the point 3 2 2 The equation 5x 6xy 5y 16 represents and ellipse dy dx a Find at 1 1 b Find two points on the ellipse at which the tangent is horizontal

Let f x 6x x 1 1 6 A Sketch the graph of f and when you re done place a 1 in the box B Find the value of x where f is discontinuous If there is no value enter NONE x values 0 values C Find the value of a where fis nondifferentiable If there is no value enter NONE

Let h x f x g x If f x 2a 2x 3 and g x x 3x 4 what is h 1

Let h x f x g x If f x 3x x 4 and g x x 2 what is h 1 Do not include h 1 in your answer For example if you found h 1 7 you would enter 7

Let f x 3 Then the expression can be written in the form A Bx Ch 3 Dx 3 where A B C and D are constants Note It s possible for one or more of these constants to be 0 Find the constants A B C D Use your answer from above to find lim h 0 lim h 0 f x h f x h Finally find each of the following f 1 f 2 3 f x h f x h f x h f x h

A thermometer reading 8 C is brought into a room with a constant temperature of 30 C If the thermometer reads 16 C after 4 minutes what will it read after being in the room for 6 minutes For minutes After 6 minutes the thermometer will read C Do not round until the final answer Then round to the nearest hundredth as needed

grams Do not round until the final answer Then round to the nearest thousandth as needed

A pizza baked at 425 F is removed from the oven at 8 00 PM and placed in a room that is a constant 72 F After 5 minutes the pizza is at 300 F a At what time can you begin eating the pizza if you want its temperature to be 130 F b Determine the time that needs to elapse before the temperature of the pizza is 190 F c What do you notice about the temperature as time passes a The temperature of the pizza will be 130 F at Use HH MM format for time Do not round until the final answer Then round to the nearest minute as needed

K Solve the equation 4 16 The solution set is Type an integer or a simplified fraction Use a comma to separate answers

K Use properties of logarithms to find the exact value of the expression Do not log 53 log 3625 log 53 log 3625 Type an integer or a simplified fraction

w the steps for graphing a rational function to graph the function H x x 125 X 64 i ne Tunction nas two odique asymptotes Ine odlique asymptote with a negative siope is Type equations Use integers or fractions for any numbers in the equations The function has no oblique asymptote A rmine the points if any at which the graph of H intersects the horizontal or oblique asymptote if one exists Select the correct choice and if necessary f ce 125 125 The graph of H intersects the horizontal or oblique asymptote at 64 64 Simplify your answer Type an ordered pair Use a comma to separate answers as needed 3 The graph of H intersects the horizontal or oblique asymptote at infinitely many points C There is no point at which the graph of H intersects the horizontal or oblique asymptote D There is no horizontal or oblique asymptote the real zeros of the numerator and denominator of H to divide the x axis into intervals Determine where the graph of H is above or below the x axis by che luating H there Select the correct choice and fill in the answer box es to complete your choice A The graph of H is below the x axis on the interval s Type your answer in interval notation Use a comma to separate answers as needed and the opique asymptote with a positive siope is B The graph of H is above the x axis on the interval s Type your answer in interval notation Use a comma to separate answers as needed C The graph of H is above the x axis on the interval s and below the x axis on the interval s Type your answers in interval notation Use a comma to separate answers as needed

x x 4 x 16 w the steps for graphing a rational function to graph the function P x 3 The function has two oblique asymptotes The oblique asymptote with a negative slope is and the oblique asymptote with a positive slope is Type equations Use integers or fractions for any numbers in the equations C The function has no oblique asymptote ermine the points if any at which the graph of P intersects the horizontal or oblique asymptote if one exists Select the correct choice and if necessary fill in the answe ce A The graph of P intersects the horizontal or oblique asymptote at Simplify your answer Type an ordered pair Use a comma to separate answers as needed 3 The graph of P intersects the horizontal or oblique asymptote at infinitely many points C There is no point at which the graph of P intersects the horizontal or oblique asymptote D There is no horizontal or oblique asymptote the real zeros of the numerator and denominator of P to divide the x axis into intervals Determine where the graph of P is above or below the x axis by choosing a nu uating P there Select the correct choice and fill in the answer box es to complete your choice A The graph of P is above the x axis on the interval s and below the x axis on the interval s Type your answers in interval notation Use a comma to separate answers as needed B The graph of P is below the x axis on the interval s Type your answer in interval notation Use a comma to separate answers as needed C The graph of P is above the x axis on the interval s Type your answer in interval notation Use a comma to separate answers as needed

K Follow the steps for graphing a rational function to graph the function R x x 6 x x 13 A The function has one horizontal asymptote y 0 Type an equation Use integers or fractions for any numbers in the equation OB The function has two horizontal asymptotes The top asymptote is and the bottom asymptote is Type equations Use integers or fractions for any numbers in the equations OC The function has no horizontal asymptote Determine the oblique asymptote s if any exist Select the correct choice and if necessary fill in the answer box es to complete your choice OA The function has one oblique asymptote Type an equation Use integers or fractions for any numbers in the equation OB The function has two oblique asymptotes The oblique asymptote with a negative slope is and the oblique asymptote with a positive slope is Type equations Use integers or fractions for any numbers in the equations C The function has no oblique asymptote Determine the points if any at which the graph of R intersects the horizontal or oblique asymptote if one exists Select the correct choice and if necessary fill in the ans choice OA The graph of R intersects the horizontal or oblique asymptote at Simplify your answer Type an ordered pair Use a comma to separate answers as needed OB The graph of R intersects the horizontal or oblique asymptote at infinitely many points OC There is no point at which the graph of R intersects the horizontal or oblique asymptote OD There is no horizontal or oblique asymptote box

3 Follow the steps for graphing a rational function to graph the function R x X 6 x x 13 The function has one vertical asymptote Type an equation Use integers or fractions for any numbers in the equation B The function has two vertical asymptotes The leftmost asymptote is x 13 and the rightmost asymptote is x 0 Type equations Use integers or fractions for any numbers in the equations C The function has three vertical asymptotes The leftmost asymptote is the middle asymptote is and the rightmost asymptote is Type equations Use integers or fractions for any numbers in the equations OD The function has no vertical asymptote Determine the hole if it exists Select the correct choice and if necessary fill in the answer box to complete your choice OA There is a hole in the graph at the point Type an ordered pair using integers or fractions B There are no holes in the graph Determine the behavior of the graph on either side of any vertical asymptotes if any exist Select the correct choice and if necessary fill in the answer box es to complete O A It approaches on one side of the asymptote s at x and on the other It approaches either co or co on both sides of the asymptote s at x Type integers or simplified fractions Use a comma to separate answers as needed Type each answer only once OB It approaches either or o on both sides of the asymptote s at x Type integers or simplified fractions Use a comma to separate answers as needed Type each answer only once OC It approaches o on one side of the asymptote s at x and co on the other Type integers or simplified fractions Use a comma to separate answers as needed Type each answer only once OD The function has no vertical asymptote

ow the steps for graphing a rational function to graph the function P x x x 4 x 16 BELLCO Lermine the role in it exisis Select the correct choice and i necessary the answer box to complete your choice A There is a hole in the graph at the point Type an ordered pair using integers or fractions B There are no holes in the graph termine the behavior of the graph on either side of any vertical asymptotes if any exist Select the correct choice and if necessary fill in A It approaches o on one side of the asymptote s at x and co on the other It approaches either co or co on both sides of th Type integers or simplified fractions Use a comma to separate answers as needed Type each answer only once B It approaches o on one side of the asymptote s at x 4 4 and o on the other Type integers or simplified fractions Use a comma to separate answers as needed Type each answer only once C It approaches either o or on both sides of the asymptote s at x Type integers or simplified fractions Use a comma to separate answers as needed Type each answer only once D The function has no vertical asymptote termine the horizontal asymptote s if any exist Select the correct choice and if necessary fill in the answer box es to complete your c A The function has one horizontal asymptote Type an equation Use integers or fractions for any numbers in the equation and the bottom asymptote is B The function has two horizontal asymptotes The top asymptie is Type equations Use integers or fractions for any numbers in the equations C The function has no horizontal asymptote

Identify any horizontal or vertical asymptotes in the graph to the right State the domain of f Identify any horizontal or vertical asymptotes in the graph Select the correct choice below and if necessary fill in the answer box es to complete your choice Type an integer or a simplified fraction Use a comma to separate answers as needed and the vertical asymptote s is are x O A The horizontal asymptote s is are y OB The horizontal asymptote s is are y OC The vertical asymptote s is are x and there is no vertical asymptote and there is no horizontal asymptote D There are no horizontal or vertical asymptotes A

K Give the equations of any vertical or horizontal asymptotes for the graph of the rational function 5 6x 2x 4 Select the correct choice below and fill in any answer boxes within your choice OA The equation of the vertical asymptote is x Type an integer or a simplified fraction Use a comma to separate answers as needed OB There is no vertical asymptote

Find the period y 2 cot 5mx Graph the function 5 2x 10 32 2 10 s j 1 t 5 10 10 5 10 5 3 10 3 2 2 2 5 10 ut fr 5 5 T 10 104 10 5 S 2 5 5 10 4 3 32 10 5

Find the period Y 3 tan 3x Graph the function M HUSAIN

As a wave passes by an offshore piling the height of the water is modeled by the function h t 8 cos t where h t is the height in feet above mean sea level at time t seconds trough crest a Find the period of the wave S b Find the wave height that is the vertical distance between the trough and the crest of the wave ft

Find the amplitude and period of the function y 4 sin x amplitude period Sketch the graph of the function y 12 9 6 3 10 5 AAMA son 12 9 6 3 6x 9 12 3 5 10 10 y 10 9 12x 5 AA 12 97 6x 3x 3x 9x 12 157 10 x 10 12 9 6 3x y 10 5 10 3A 9x

Graph the function h x 3 cos x y 0 0 34 X m nu 2A

Graph the function f x 5 sin x O GL y 41 O 41 2

9 Using first principles determine the equation of the tangent line at point 1 13 on the curve f x x 5x 7 5 mark

Find the slope of the tangent line at point 1 2 on the curve f x 5x 2x using First Principles 4 marks

Evaluate the indefinite integral Use C for the constant of integration sin t 1 cos t dt 3 2 cos 1 1 cos 1 3 sin 1 C

Use a graphing utility to graph the function Use the graph to determine the behavior of the function as x c f x sec x a x f x b x f x f x TL f x 2 T 2 as x approaches from the right right 2 2 TU d x as x approaches from the left 2 TU as as x approaches from the right 2 as x approaches from the left

Match the function with its graph g x x cos x 2 0 2 4 6 2 2 2 my m 2 4 O 6 4 2 4 5 2 5 4 MM 2T 0 2 02 27 Describe the behavior of the function as x approaches zero As x 0 f x 2 4 5 5 02 0 2 2T 2T

Use a graph to solve the equation on the interval 2x 2r List the solutions in increasing order from left to right on the x axis csc x 2 3 3 x X X X smallest value largest value

Use the graph of the function to answer each question 2x sxs 20 f x cot x a Find all x intercepts of the graph of y fx Enter your answers as a comma separated list If an answer does not exist enter DNE b Find all y intercepts of the graph of y fx Enter your answers as a comma separated list If an answer does not exist enter DNE yu decreasing c Find the intervals on which the graph of y fx is increasing and the intervals on which the graph of y f x is decreasing Enter your answers using interval notation If an answer does not exis increasing d Find all relative extrema if any of the graph of yfx Enter your answers as a comma separated list of ordered pairs If an answer does not exist enter ONE relative minima relative maxima MY NOTES e Find all vertical asymptotes if any of the graph of y f Enter your answers as a comma separated list If an answer does not exist enter DNE A

Use the graph of the function to determine whether the function is even odd or neither f x sec x 2 The function is even O The function is odd O The function is neither even nor odd

Question 7 10 points A metal sphere is placed in seawater to study the corrosive effect of seawater If the surface area decreases at 35 cm year due to corrosion how fast is the radius changing when it is 12 cm Paragraph V B I U A Stm 0 v X

An engine cylinder 15 0 cm deep is being bored such that the radius increases by 0 100 mm min How fast is the volume V of the cylinder changing when the diameter is 9 50 cm Paragraph V BIU A v v I On v

For the function F x oblique asymptotes Ho a graph the rational function using transformations b use the final graph to find the domain and range and c use the final grap x 4 b The domain of the given function is 4 U 4 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression The range of the given function is 0 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression c What is are the vertical asymptote s Select the correct choice and if necessary fill in the answer box es to complete your choice There is one vertical asymptote It is x 4 Type an equation Use integers or fractions for any numbers in the equation B The left vertical asymptote is The right vertical asymptote is Type equations Use integers or fractions for any numbers in the equations COC OC There is no vertical asymptote What is are the horizontal asymptote s Select the correct choice and if necessary fill in the answer box es to complete your chce OA The top horizontal asymptote is The bottom horizontal asymptote is Type equations Use integers or fractions for any numbers in the equations OB There is one horizontal asymptote It is Type an equation Use integers or fractions for any numbers in the equation OC There is no horizontal asymptote E

x 1 Follow the steps for graphing a rational function to graph the function H x x 64 125 If needed first write the given function as a single rational expression Then factor the numerator and denominator of H x Select the correct choice complete your choice A H X x 5 x 5x 25 x 8 x 8 OB H x cannot be factored further What is the domain of H x 8 U 8 8 U 8 00 Simplify your answer Type your answer in interval notation Use integers or fractions for any numbers in the expression Write H x in lowest terms Select the correct choice and if necessary fill in the answer box to complete your choice COLLE Type your answer in factored form Do not simplify O A H x Simplify your answer OB H x is already in lowest terms

For the function F x oblique asymptotes A 1 Shift the graph of y 2 X 1 a graph the rational function using transformations b use the final graph to find the domain and range and c use a x 1 OB Shift the graph of y Choose the correct graph O A XN 1 D Shift the graph of y X Ay 2 1 C Shift the graph of y 1 unit left x 2 1 unit up 2 1 unit down 1 unit right b The domain of the given function is B Ay Q Q station Lice inteners or fractions for any numbers in the expression www O C 10 Q Q D G

oblique asymptotes Choose the correct graph O A TOT 10 10 O B 10 110 b The domain of the given function is 3 U 3 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression OA There is one vertical asymptote It is Type an equation Use integers or fractions for any numbers in the equation OB The left vertical asymptote is The right vertical asymptote is frontions for any numbers in the equations O C 19 The range of the given function is 0 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression c What is are the vertical asymptote s Select the correct choice and if necessary fill in the answer box es to complete your choice 10 and c use the final graph to lis D Ay Q G

Solve the inequality algebraically x 20 x 20 X SO Interval Sign Move all terms to the left side of the inequality and then define the left side of the inequality as f x Then list the intervals determined by the real zeros undefined Complete the following table Type your answers in interval notation Use ascending order E

f x 2x11x 11x 15x 9 Make sure to explain how you determined the significant characteristics of the function Those characteristics should include end behavior and x and y intercepts Answer Sel 40 30 20 10

Graph the function f x 6 x in the standard viewing window of your calculator and trace from left to right along a representative portion of the interval o o Then fill in the blank of the following sentence with either increasing or decreasing Over the interval o the function f x 6 x is Graph the function f x 6 x Choose the correct graph below Use the viewing window 5 5 1 15 15 1 A B A O C HE 24 G OD

A person is driving a car on a straight road The graph shows the distance y in miles that the individual is from home after x hours Find the slope intercept form of the line y 0x 0 COP DLSLAACG LELO Ay 160 m 80 0 0 1 50 2 3 120 OU Time in hours

Through 6 7 perpendicular to 6x y 4 O The equation of the line is y Simplify your answer Type your answer in slope intercept form Use integers or fractions for any numbers in the expression

K Find the slope intercept form of the equation of the line satisfying the given conditions Do not use a calcul Through 5 25 and 0 5 The slope intercept form of the line is y Simplify your answer Use integers or fractions for any numbers in the equation Do not factor

Graph the following line Also give the a x intercept if any b y intercept if any c domain d range and e slope of the line if defined Do not use a calculator X 5 a Find the x intercept Select the correct choice below and if necessary fill in the answer box to complete your choice A The x intercept is 5 0 Type an ordered pair using integers or decimals OB There is no x intercept C b Find the y intercept Select the correct choice below and if necessary fill in the answer box to complete your choice A The y intercept is Type an ordered pair using integers or decimals B There is no y intercept c Find the domain Select the correct choice below and fill in the answer box to complete your choice OA The domain is Type your answer in interval notation OB The domain is Type an integer or a decimal Use a comma to separate answers as needed Ay

Graph the following linear function Give the a x intercept b y intercept c domain d range and e slope of the line Do not use a calculator f x x 9 c Find the domain of the function Select the correct choice below and fill in the answer box to complete your choice A The domain is 00 00 Type your answer in interval notation The domain is Type an integer or a decimal Use a comma to separate answers as needed d Find the range of the function Select the correct choice below and fill in the answer box to complete your choice OA The range is Type an integer or a decimal Use a comma to separate answers as needed B The range is 00 00 Type your answer in interval notation e Find the slope of the line f x x 9 Select the correct choice below and if necessary fill in the answer box to complete your choice A The slope of the line f x x 9 is OB The slope is undefined

Determine the domain and range of the relation and tell whether the relation is a function COLLE Using interval notation the domain is 4 4 Simplify your answers Type your answer in interval notation Using interval notation the range is Simplify your answers Type your answer in interval notation

Use the graph of y f x to find each function value b f 0 a f 3 c f 5 d f 6 a f 3 0 b f 0

Evaluate the integral 59 sec 460 de

Find the equations of any vertical asymptotes for the function below x2 5x f x 3 x 36x Determine the equation of any vertical asymptotes Select the correct choice below and if necessary fill in the answer box within your choice OA The vertical asymptote s is are x Simplify your answer Use a comma to separate answers as needed OB There are no vertical asymptotes

Express the integrand as a sum of partial fractions and evaluate the integral OB X 4 x 9x 10 O C Express the integrand as a sum of partial fractions Select the correct choice below and fill in any answer boxes to complete your choice Use integers or fractions for any numbers in the expression O A X 4 x 9x 10 S dx S x 4 SZ dx x 9x 10 X 4 dx x 9x 10 S dx S W x 10 X 1 X 4 X 4 x 10 dx dx 2 X dx