Calculus Questions

Find dy dx by implicit differentiation 2x 5xy y 5 y

Differentiate f v f v V v d

Suppose that f 5 1 f 5 4 g 5 6 and g 5 9 Find the following values a fg 5 b f g 5 c g f 5

Find the derivative of f x 1 9x x x in two ways Use the Product Rule f x Perform the multiplication first f x Do your answers agree Yes O No

Differentiate y y 1 seca tan a

If H x x cos x find H x and H x H x H x

Differentiate y y t t 1

Differentiate f x f x sec 2 8 secx

Differentiate y y 26 3 25

Differentiate y y x 6 x x 2

Differentiate f t f t 4t 4 t

59 A retailer has been selling 1200 tablet computers a week at 350 each The marketing department estimates that an addi tional 80 tablets will sell each week for every 10 that the price is lowered a Find the demand function b What should the price be set at in order to maximize revenue

29 Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r

34 sin 450 Convert each degree measure into radians 35 300 37 100 Convert each radian measure into degrees 36 930 201

3 Inverse Values 3 pages max 10 pts Find all possible complex values associated with w 2 sin z at z 3 using the quadratic solve approach Identify which of these values corresponds to the value obtained from wolfram a Evaluate the derivative w 3 Do the same for the value of tanh 3i

2 Use the limit comparison test to determine if 2 3 n converges or diverges

1 Use the comparison test to determine if 8 n 1 n 1 n 2 converges or diverges

A thermometer is taken from a room where the temperature is 21 C to the outdoors where the temperature is 13 C After one minute the thermometer reads 10 C a What will the reading on the thermometer be after 5 more minutes b When will the thermometer read 12 C minutes after it was taken to the outdoors

Use the integral test to determine if the series converge or diverge 1 0 2 n 1 n a b 1 n 2 n ln n

8 4 points Determine ALGEBRAICALLY if the function f x 3x 8 is even odd or neither

A graphing calculator is recommended Use a graphing calculator to evaluate the definite integral rounding your answer to three decimal places Hint Use a command like FnInt or ff x dx 471 Lets d dx X

oblem 2 When a golfer strikes a golf ball and we observe the ball s trajectory we can represent its vertical height h measured in feet above the ground after t seconds using the following representation h t 96t 16t a Find the height of the ball after 4 seconds Is the golf ball moving upward or downward after 4 seconds b Evaluate the rate of change of h t on the time interval 3 3 5 Interpret the meaning of your answer and be sure to include appropriate units c Find the maximum height of the golf ball d State a realistic domain for the function h t e Sketch the function h t and label when the ball is moving upward moving downward and the point where the golf ball is at its highest point include the coordinates of the highest point

blem 5 For the rational function R x a State the domain of R x b Find all zeros 3x 6 x 3 C find c Find all intercepts d Find all poles and identify all vertical asymptotes e Generate a sign chart to determine where R x is positive and where it is nega f Identify all horizontal asymptotes Sketch the monk of DU

y f x 89876 32 10 5 y 10 9 8 7 6 5 4 3 2 1 2 234567O9O 3 4 5 6 7 8 9 10 Evaluate g f 2 1 2 3 4 5 6 8 9 10 y g x 10 8 7 6 5 43 2 10 999664321 36789 W Ni 2 3 4 5 6 8 9 10 1 2 3 4 5 6 7 8 9 10

5 f r x f x dx for the function f x graphed below Hint No calculus is necessary just as we used integrals to find areas we can use areas to find integrals The curve shown is a quarter circle Find 53 y 24 1 2 square units 3 4 5

H t 10 9 8 6 5 4 if 1 0 if t 0 dom rang

Determine whether the series n 0 O B converges or diverges If it converges find its sum Select the correct choice below and if necessary fill in the answer box within your choice OA The series diverges because lim n The series converges because lim n 8 T n T 0 or fails to exist n 0 The sum of the series is Type an exact answer O c The series diverges because it is a geometric series with 21 O D The series converges because it is a geometric series with r 1 The sum of the series is Type an exact answer

EXAMPLE 2 Velocity and acceleration Suppose the position in feet of an object moving horizontally along a line at time t in seconds is s 51 for 0 1 5 S Figure 3 42 2 2 1 2 3 4 5 Position s 2 5t

2 For the quadratic function f x x 7x 10 a Find the vertex b Does the vertex represent a local maximum or a local minimum c State the axis of symmetry d Find the y intercept

3 Use your calculator to graph f x x x 4x x 5 Then find the following Round 10 pts answers to 2 decimal places as needed a Local maxima stated as ordered pairs b Local minima stated as ordered pairs c Increasing intervals d Decreasing intervals dy intercepts stated as ordered pairs

5 Solve the inequality State your answer in interval notation 5r 2 0

3 Use a graphing utility to approximate the real solutions Round to two decimal places Son f x x 4x 2

77 4x 5y 5 9 1 54321 9 2x 5y 0 6 15 s 4 3 2 1 11 y x 5 54321 y Ay y 23 X 4 X x 8 10 12 8 6x y 4 6 5 4 3 2 1 10 y x 2 6 5 4 3 41 12 y2 3x 1 6543 2 AY Ay Ay 43 15 x 15 X

The price p in dollars and the quantity x sold of a certain product satisfy the demand equation x 5p 200 Answ parts a through g ww a Find a model that expresses the revenue R as a function of p Remember R xp R p Simplify your answer Use integers or decimals for any numbers in the expression

the graph of the quadratic function f to determine the solution a Solve f x 0 b Solve f x 0 a The solution to f x 0 is Type your answer in interval notation

2 12 pts Use long division to divide P x by D x and express P in the form P x D x Q x R x Note that this answer is in a different form than that of Problem 1 P x 8x 4x 6 D x 2x 1

A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function R x 100x 0 2x where the revenue R x is measured in dollars What is the maximum revenue and how many units should be manufactured to obtain this maximum at units

Carol has 1100 ft of fencing to fence in a rectangular horse corral 550 x a Find a function that models the area A of the corral in terms of the width x of the corral A x b Find the dimensions of the rectangle that maximize the area of the corral width ft ft length

The effectiveness of a television commercial depends on how many times a viewer watches it After some experiments an advertising agency found that if the effectiveness E is measured on a scale of 0 to 10 the E n 3n 34 where n is the number of times a viewer watches a given commercial For a commercial to have maximum effectiveness how many times should a viewer watch it times

If a ball is thrown directly upward with a velocity of 24 ft s its height in feet after t seconds is given by y 24t 16t2 What is the maximum height attained by the ball Round your answer to the nearest whole number Need Help Read It Watch It Manteell

Find the maximum or minimum value of the function g x 100x 1500x Is this a maximum or minimum value O maximum value minimum value

The graph of a quadratic function f is given f x 4x 8x 1 vertex x intercepts y intercept 5 a Find the coordinates of the vertex and the x and y intercepts Round you x y x y x y range 1 x y 5 b Find the maximum or minimum value of f The Select value of f is f x X smaller x value larger x value c Find the domain and range of f Enter your answers using interval notation domain

quadratic function f is given f x x 14x 35 a Express fin standard form f x b Sketch a graph of f 60 40 20 O 60 40 20 y 60 40 20 20 40 60 y 60 40 20 20 40 60 20 Is this a maximum or minimum value O maximum value minimum value 20 c Find the maximum or minimum value of f f x 40 40 60 60 X X 60 40 60 40 20 20 20 40 60 40 20 40 20 60 20 60 40 60 y y

Find the maximum or minimum value of the function f x x 2x 5 Is this a maximum or minimum value maximum value O minimum value

The graph of f x 5 x 9 2 2 is a parabola that opens downward with its vertex at x y and f 9 is the maximum value of f

The quadratic function f x a x h k is in standard form a The graph of f is a parabola with vertex x y b If a 0 the graph of f opens upward c If a 0 the graph of f opens downward Need Help Read It In this case f h k is the minimum In this case f h k is the maximum v value of f value of f

10 0 y 5 10 b Find the horizontal asymptote Enter None if appropriate 5 10 a Find the vertical asymptote s Separate multiple answers with commas Enter None if appropriate powered by desmos

Progress Values of f t are given in the following table The function appears to have a 2 4 6 8 10 f 1 138 134 127 119 106 91 a Does this function appear to have a positive or negative first derivative Second derivative first derivative and a 2 i 0 b Estimate 2 and f 8 Use difference quotients to the right second derivative

5 So far we have been using an artificial example because we could calculate the actual value of 5x 3x 1 dx just by using the Fundamental theorem of calculus and finding an antiderivative Now we use a more motivated example 2 a Now consider 1 9x dx Does this function have an 0 elementary antiderivative i Answer ii Note this integral arises when trying to find the length of 3 the curve y x as x goes from 0 to 2 b Now try to estimate the integral of that function from a 0 to b 2 using Riemann sums then record how many subintervals you needed to convince you that your error is less than 001 In parts b c d you will have to experiment with values of n until you are convinced your answer is within 0 001 of the actual value i Answers c Repeat this process but using the trapezoidal rule now How many subintervals were necessary to reach this level of accuracy i Answer d Finally do this again with Simpson s rule but remember to only use even values of n for Simpson s rule i Answer e Of these methods which one converged the fastest Did this result surprise you Share your thoughts i Answer f In each of these cases you also did not have a means of knowing the exact value of the integral so how did you determine when your error was less than 001 Answer

Suppose that f t is a function with f 25 3 8 and f 25 0 5 Estimate f 26 and f 28 f 26 28