Calculus Questions

Find the slope intercept form of the line satisfying the given conditi X 4 20 2 y 27 17 7 3 13 O A y 5x 7 OB y 10x 13 OC y 10x 13 P Y 5x 7

Which is the definition of the derivative of f x 6x 6 x h 6 h f x lim h 0 O f x 6 f x lim h 0 f x lim 6 x h 6hx h f x lim h 0 6x h h 0 h 6 x h 6x h

Current Attempt in Progress Which is the definition of the derivative of f x f x lim h 0 f x lim h 0 x h h 7 x h 7hx f x lim h 0 h 7 x h 7x h f x lim h 0 Of x 21x 7 xh 7hx h 7x

The position of an object moving on a line is given by s t 6t t t2 0 where s is in metres and t is in seconds a b c Determine a model to represent the velocity Sketch the model for velocity Determine the velocity at t 2

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 h x f x g x If f 6 4 f 6 2 g 6 2 and g 6 5 what is h 6 Do not include h 6 in your answer For example if you found h 6 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

Let f x 4 x 8 f x lim h 0 Your answer above and the next few answers below will involve the variables h and z The expression inside the limit simplifies to a simple fraction with Numerator Then according to the definition of derivative Denominator NOTE Both of the previous two parts must be correct in order to receive partial credit We can cancel the factor appearing in the denominator against a similar factor appearing in the numerator leaving a simpler fraction with Numerator Denominator NOTE All three of the previous parts must be correct to receive partial credit Taking the limit of this fractional expression as h 0 gives us f x

The graph of f is shown b d Evaluate each integral by interpreting it in terms of areas a f x dx e 64 20 160 y f x A 8 16 24 48 28 20 RX f x dx 132 8 28 FX 12 36 f x dx 28 12 1Rx f x dx If x dx f f x dx Xx 32 x

The graph of f is shown b c d e f 64 Evaluate each integral by interpreting it in terms of areas a Rx dx 20 160 48 8 28 0 F f x dx 28 f x dx f x dx 2 1Rx dx f x dx 8 x y f x 16 24 32 X

Graph the piecewise function to find its domain and range Your answers must be in interval notation Domain Range help intervals help intervals f x x 1 4 2x if x 3 if x 3

Graph the piecewise function to find its domain and if x 3 if x 4 f x 2x 1 x Domain Range range help intervals help intervals

Evaluate the definite integral 12 2 110 6 X 1 2x dx

18 10 140 10 10

Find the domain of f x inf 5 U 1 inf U 1 Answers Attempt 4 of 4 0 9x 9 inf A help intervals Answer Enter the solution in interval notation 6

Find the domain of f x 8 x 1 Enter the solution in interval notation A help intervals x 1

For f x x 6 and g x x 4 find the following composite functions and state the domain of each a fog b gof c fof d gog a fog x Simplify your answer

8x The function f x is one to one 9x 1 a Find its inverse function f1 and check your answer b Find the domain and the range of f and f a f x Simplify your answer

Use the definition of the derivative don t be tempted to take shortcuts to find the derivative of the function f x 3x 4 Then state the domain of the function and the domain of the derivative Note When entering interval notation in WeBWork use I for I for and U for the union symbol If the set is empty enter without the quotation marks x Domain of f x Domain of f x

Suppose that Find f x f x f x h f x 5hx 7h 6

Let f x x 41 Evaluate the following limits lim 4 1 2 f 4 lim 4 f 2 1 4 Thus the function f x is not differentiablo at 1

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

tion list estion 5 estion 6 uestion 7 uestion 8 Question 9 Question 10 K e11x 18 The solution set is Type an integer or a simplified fraction Use a comma to separate an

K Find the domain of the function g x In x 2 The domain of g is Type your answer in interval notat

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

K ange of Base Formula and a calculator to evaluate the logarithm log 742 log 42 Do not round until the final answer Then round to three decimal places as ne

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

llow the steps for graphing a rational function to graph the function R x 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 D There is no horizontal or oblique asymptote Use the real zeros of the numerator and denominator of R to divide the x axis into intervals Determine where the graph of R is above or b evaluating R there Select the correct choice and fill in the answer box es to complete your choice A The graph of R is above the x axis on the interval s 13 6 0 and below the x axis on the interval s 13 6 0 Type your answers in interval notation Use a comma to separate answers as needed x 6 X X 13 OB The graph of R is above the x axis on the interval s Type your answer in interval notation Use a comma to separate answers as needed OC The graph of R is below the x axis on the interval s Type your answer in interval notation Use a comma to separate answers as needed Use the results from the previous steps to graph R Choose the correct graph SOA Q O B Q O C

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

x 125 64 Follow the steps for graphing a rational function to graph the function H x OA The function has one horizontal asymptote Type integers or simplified fractions Use a comma to separate answers as needed Type each answer only once B It approaches either o or o on both sides of the asymptote s at x Type integer or simplified fraction Use a comma to separate answers as needed Type each answer only once C It approaches o on one side of the asymptote s at x 8 8 and o on the other Type integer or simplified fraction Use a comma to separate answers as needed Type each answer only once OD There is no vertical asymptote Determine the horizontal asymptote s if any exist Select the correct choice and if necessary fill in the answer box es to complete your choice 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 C 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 OC The function has no oblique asymptote

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

13 y x 3 1 x

lve the initial value problem h x 0 2 t dx dt 2a

b Use a calculator or computer to find Lexd Round your answers to three decimal places dx i e x6 dx i L e dx er Led i dx i affe eTextbook and Media c The improper integral e to dx for a 1 a 2 a 3 a 7 Round your answer to three decimal places 4 e x dx converges to a finite value Use your answers from part b to estimate that va

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

K 1 Use transformations of the graph of either f x g x O A 3 x 3 Choose the correct graph below Q Q or h x OB 1 to sketch a graph of y g x by hand Show all asymptotes Write g x in terms of either f x or h x O C O D B

nsider the all equation dy y y 1 y 2 y 0 0 dt a 2 points Find the equilibria b 4 points Draw the phase line diagram and determine stability of equilibria c 7 points Find the general solution to the differential equation Notice that it is separable

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

5 2 points A type of ocean wave found near the equator called a Kelvin wave is described by the following equations du h g 0 t x du x h t H 0 Here h h x t is the height of water u and 9 and H are constants representing gravity and average water height u x t is the water s east west velocity 0 h a Eliminate u from the system of equations to find 32 gHh 0 Hint use Clairaut s Theorem b Solutions to equations involving waves typically look like Acos kx wt The coefficient w is a time frequency of the wave and k describes the wave s spatial frequency Treat A as a constant and substitute this solution into the equation you found in a What is the relationship between w and k c Assume k and w are positive We describe the velocity of waves in two ways The crest of a wave moves with phase speed A group of waves moves with group velocity Show that the phase speed and group velocity of a Kelvin wave are the same We call this type of wave nondispersive which means that it travels without changing shape Kelvin waves take about 2 months to go across the Pacific Ocean

Graphy 22 12 x2 and y Select each correct answer on the same axes Which do you think is larger x3 1 dx is greater because the area under y is greater than the area under the graph of y S 23 1 1 6 12 1 2 dx is greater because the area under y dx is greater because the area under y S dx is greater because the graph of y 1 3 dx ordx Why dx is greater because the graph of y dx is greater because the graph of y 1 1 is greater than the area under the graph of y x3 1 is less than the area under the graph of y x 1 is above greater than the graph of y x2 1 is below less than the graph of y is above greater than the graph of y x3 x x2

Find the period y tan 5x Graph the function 10 15 Sn H 4 n 5 5 S 10 10 5 2 Sx 2 1 5 15 3 3 S 10 X 24 2x 3x 10 14 HA A 10 5 101 10 S 10 A 10 32 ff