Application of derivatives Questions and Answers

Question Determine the exact values of sin 0 and cos 0 given that the terminal side of angle 0 intersects the unit circle in the first quadrant at Provide your answer below

Question Find an angle A that is coterminal with an angle in radians measuring 23 where 0 A 2 Give your answer as an exact answer involving if necessary

A student takes a multiple choice test of 20 with four choices and guesses on each question Find the probability that the student gets exactly 6 correct answers 0 1686 O 0 3819 0 0745

Identify the x intercepts of the parabola y x 1 16 O 3 0 and 5 0 O 2 0 and 4 0 O 0 3 and 0 6 O 3 0 and 5 0

Solve the exponential equation 37 3 243 Ox 3 Ox 2 3 x 2 3 Ox 4

Find the vertical and horizontal asymptote s of the graph of the rational function f x 2x 3 x x 6 O Vertical Asymptotes x 1 x 6 Horizontal Asymptote x axis Vertical Asymptotes x 2 x 3 Horizontal Asymptote x axis O Vertical Asymptotes x 2 x 3 Horizontal Asymptote x axis O Vertical Asymptotes x 2 x 3 Horizontal Asymptote y 2

In a club with 8 members how many ways can the club elect a president a treasurer and a secretary 358 O 112 56 336 240

Solve sin x 0 75 on 0 x 2 There are two solutions A and B with A B and A and B in radians to 4 decimal places A B

Let f 2 423 1922 the remaining zeros of f Oax 2 3i and x 3 4 b x 2 3i and x 3 4 r Oca 2 3i and x 2 3i Odx 2 3i and x 4 3 Oex 2 3i and x 3 4 x 2 3i and x 3 4 O 642 39 If it is known that r 2 3i is a zero of f find Of

a x y Solve the system u 2x y v 2x 2y for x and y in terms of u and v Then find the value of the Jacobian Find the image under the transformation of the triangular region with vertices 0 0 2 4 and 2 2 in the xy plane Sketch the transformed region in the uv plane a u v The function for x in terms of u and vis x The function for y in terms of u and v is y The Jacobian of the transformation is J u v 6 0 2u v 6 Av V U 3 Choose the correct sketch of the transformed region in the uv plane below A 0 12 1 6 Q 5 B 0 12 6 0 Q Q GALLE OC 0 12 6 0 Q 9 G OD AV 6 0 0 12 od u

me integrals below give the area of a region in the xy plane Sketch the region label each bounding curve with its equation and give the coordinates of the points where the curves intersect Then find the area of the region TC 4 3 cos x S S 03 sin x dy dx Sketch the region label each bounding curve with its equation and give the coordinates of the points where the curves intersect Choose the correct answer below A OB VES sinx V3 cos x E The area of the region is 3 2 1 Type an exact answer using radicals as needed Ay y 3 cos x 1 3 sin x Q 3x O C x 312 42 y 3 cos x y 3 sin X OU 2

ketch the region of integration associated with the double integral 2 SS f x y dy dx 0x Choose the correct sketch of the described region OA 5 OB 8 13 O OU OD

Solve the system u 2x y v 2x 2y for x and y in terms of u and v Then find the value of the Jacobian The function for x in terms of u and v is x The function for y in terms of u and v is y The Jacobian of the transformation is J u v Choose the correct sketch of the transformed region in the uv plane below O A OB 6 0 0 12 Q Q 0 12 6 0 xx y Find the image under the transformation of the triangular region with vertices 0 0 2 4 and 2 2 in the xy plane Sketch the transformed region in the uv plane u v Q GITS OC 0 12 6 0 Q Q OD U 6 0 0 12 O a

equations for a the tangent plane and b the normal line at the point Po 1 1 0 on the surface x y z 2 Which equation describes the tangent plane OA x y z 2 0 OB x y z 2 0 OC x y z 2 0 OD x y z 2 0 b Select the equations for the normal line OA x 1 t y 1 t z t OC x 1 t y 1 t z t OB x 1 t y 1 t z t OD x 1 t y 1 t z t

Given the parametric equations x t 2 and y t 3 4t where 3 t 3 which of the following graphs represents the curve and its orientation 8 7

To find J x sin 4x2 cos 9x2 dx we can use formula 81 shown below cos a b u cos a b u 2 a b 2 a b sin au cos bu du Using this we have u a C and b

Write a slope intercept equation for a line that passes through 7 0 and 5 3 The slope intercept equation for the line is Simplify your answer Use integers or fractions for any numbers in the equation

Ex 13 The order is for Zithromax PO 500 mg on day 1 then 250 mg qd on days 2 5 for a total dos of 1 5 g The label shows 200mg per 5 mL a How many should be given on day 1 b How many should be given on days 2 5 per day c Would the 30 ml size be large enough for the full course

25 Suppose that points A R and V are collinear If AR 7 and RV 5 then which point cannot possibly lie between the other two

Substitution then gives us e98 sin 80 de C

Evaluate the definite integral 9 t4 In 2t dt 3 000 or 1170

Given the series n 1 n 1 5 Does this series converge or diverge O diverges O converges If the series converges find the sup

Applying formula 69 now gives us the following Remember to use absolute values where appropriate Ita tan 16 x dx C

8 Suppose a population of bacteria is changing at a rate of dp 2 500 dt 1 0 25t where t is the time in days The initial population when t 0 is 2 000 Write an equation that gives the population at any time t a p t 10 0001n 1 0 25t 1 000 b p t ln 1 0 25t 2 000 c p t 2 500ln 1 0 25t 2 000 d p t 2 5001n 1 0 25t 1 000 e p t 10 0001n 1 0 25t 2 000

A graph of y f x is shown In the same coordinate system sketch a graph of the derivative function y f x over the interval 4 4 4 3 2 1 4 3 2 1 1 2 3 2 3 Clear All Draw Line Segment Hint Start by finding where the derivative is equal to zero Then estimate the value of the derivative two units to the left and two units to the right of the vertex of the Make sure the height of your graph matches those values

The graph of x 9y 5 0 crosses the y axis at y

Suppose we declare the following 4 byte int int x 1122334455 and store this in memory location 0x100 on a little endian system What values would be stored in the following memory locations a 0x100 b 0x101 c 0x102 d 0x103

Describe the given region in polar coordinates such that 0 r and 0 0 2 The lower portion of the region is 0 rs The upper portion of the region is m 3

If the terminal side of angle A goes through the point 25 V5 on the unit circle then what is cos A Provide your answer below

Consider the following polynomial function Step 2 of 2 Sketch the graph by identifying where the graph is positive and where the graph is negative Answer r x x 2x 8 Reset Select the area above the axis to plot the positive curve for that region and select the area below the axis to plot the negative curve for that region Use the remove all curves 10 45 A5 15 K Enable Zoom Pan

Find an angle A that is coterminal with an angle in radians measuring where 0 A 2 Give your answer as a exact answer involving if necessary Jour

If the terminal side of angle 0 passes through a point on the unit circle in the first quadrant where x 0 what is the exac measure of 0 Provide your aprwer holow

Describe the given region in polar coordinates such that 0srand 0 0 2 0s srs 0 Type exact answers using T S Poded 10 10 10 10

Write out the form of the partial fraction expansion of the function Do not determine the numerical values of the coefficien 2x a x 2 3x 1 b X x3 14x 49x X

1 point f x 7x 21x 21x 8x 2x 4 x 3x 6x 8x 2 What is the equation of the horizontal asympotote y Does the graph of f x intersect its horizontal asymptote yes or no At what x values does f x intersect its horizontal asymptote Give your answers in increasing order If f x does not intersect its horizontal asymptote leave this question blank

Find the values of and at the given point dy 2z 5xy 5yz 3y 19 0 2 1 2 2 1 2 Type an integer or a simplified fraction

For the given functions a express dw dt w z sin xy x 2t y ln t 4 z e 5 t 5 Then b evaluate dw as a function of t both by using the Chain Rule and by expressing w in terms of t and differentiating directly with respect dt a Express as a function of t dw dt dw at the given value of t

The given central difference table is generated from a polynomial Which of the following is true about this polynomial X 0 0 2 0 4 0 6 0 8 1 1 2 1 4 f x 0 0 92 1 36 0 84 1 12 5 11 28 20 44 8y 1 2 8 y 0 92 0 44 0 52 1 96 3 88 6 28 9 16 0 48 0 96 1 44 1 92 2 4 2 88 8 y 1 2 0 48 0 48 0 48 0 48 0 48 84y 0 0 0 0 polynomial degree 4 coefficient of lowest degree term 0 polynomial degree 3 coefficient of the highest degree term 10 C polynomial degree 4 coefficient of the highest order term 10 polynomial degree 3 coefficient of lowest degree term 0 48

3 Solve by completing the square x 6x 77 0 4 Solve the following

4 Use the following information to calculate MST x 8 86 X 7 96 s 4 24 s 3 86 s 3 5 6 Round your answer to the nearest thousandths Answer X3 8 40 n 18 n 16 n3 20

Use the function below to answer parts a and b 19 f x 4x 2 5x on 2 2 a Find the critical points of the function on the given interval The critical point s occur at x Simplify your answer Use a comma to separate answers as needed Type an exact answer using radicals as needed b Use a graphing utility to determine whether each critical point corresponds to a local maximum local minimum or neither Select the correct choice below if necessary fill in the answer box to complete your choice A 15 The critical point s x correspond s to the local maximum maxima Simplify your answer Use a comma to separate answers as needed Type an exact answer using radicals as needed OB There is no local maximum Select the correct choice below and if necessary fill in the answer box to complete your choice A to the local minimum minima

Find the intervals on which f is increasing and decreasing Superimpose the graphs of f and f to verify your work f x x 4x Select the correct choice below and if necessary fill in the answer box es within your choice OA The function is increasing on and decreasing on Simplify your answers Type your answers in interval notation Use a comma to separate answers as needed OB 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 C The function is decreasing on 0 00 The function is never increasing Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed OD The function is never increasing nor decreasing

Find positive numbers x and y satisfying the equation xy 25 such that the sum 5x y is as small as possib Let S be the given sum What is the objective function in terms of one number x S 5x 25 X Type an expression The interval of interest of the objective function is 0 0 Simplify your answer Type your answer in interval notation The numbers are x 5 and y 5 5

Sketch a graph of a function f that is continuous on 00 00 and has the following properties f x 0 and f x 0 on 5 f x 0 and f x 0 on 5 00 Choose the correct answer below OA Ay O B O D

A long jumper leaves the ground at an angle of 22 above the horizontal at a speed of 11 m s The height of the jumper can be modeled by x 0 053x 0 364x where it is the jumper s height in meters and x is the horizontal distance from the point of launch Part 0 3 Part 1 of 3 a At what horizontal distance from the point of launch does the maximum height occur Round to 2 decimal places The long jumper reaches a maximum height when the horizontal distance from the point of launch is approximately meters

Rewrite the following as one logarithm where the coefficient is 1 1 In x 4 4ln x In 16x 2

One way to model exponential growth and decay is with percentages Doing so we can use the following equation where a is the starting value r and its sign is the growth decay rate as a decimal and t is the time A a 1 r Note that the time and percent need to have the same respective units of time a Vehicles are known to depreciate at a rate of approximately 16 per year Write a model for the depreciation of a brand new 35 000 vehicle over time b In what year will your vehicle be worth 10 000

Expand the following logarithm as much as possible log x y

A scientist begins with 250 grams of a radioactive substance After 340 minutes the sample has decayed to 32 grams a Rounding to five decimal places write an exponential equation representing this situation b To the nearest minute what is the half life of this substance

4 2 X 2 2 X y f x 6 8 a On what interval s is f increasing Enter your answer in interval notation x On what interval s is f decreasing Enter your answer in interval notation b At what value s of x does f have a local maximum Enter your answers as a comma separated list At what value s of x does f have a local minimum Enter your answers as a comma separated list c On what interval s is f concave upward Enter your answer in interval notation