Application of derivatives Questions and Answers

Find an equation of the tangent line at x 0 y x cos x Use symbolic notation and fractions where needed equation

Calculate the following derivative Use symbolic notation and fractions where needed Kaminubleth d dx 7e 4x 2e 10x 10

Compute the derivative y cos te Express numbers in exact form Use symbolic notation and fractions where needed POLECEN

Find the derivative of y using the appropriate rule or combination of rules y x cos x Express numbers in exact form Use symbolic notation and fractions where needed dy dx

T Find an equation of the tangent line to f x 7 cos x at x 3 Let y f x and express the equation of the tangent line in terms of y and x equation

Find the indicated derivatives using the following information f 9 1 f 9 1 g 9 5 g 9 3 fg 9 2 f 9 9 0 08 X

If f x 2x 5x 3 find f 4 21 Use this to find the equation of the tangent line to the parabola y 2x 5x 3 at the poi 4 55 The equation of this tangent line can be written in the form y mx b where m is 21 X and where b is 139 X

A large koi pond is filled from a garden hose at the rate of 14 gal min Initially the pond contains 400 gal of water a Find a linear function V that models the volume of water in the pond at any time t V t b If the pond has a capacity of 1604 gal how long does it take to completely fill the pond min

Compute the curvature of the curve defined below at the stated point r t ti j t k t 1

7 Compute the curvature of the curve defined below at the stated point k 11

9 Consider the curve defined below Find T t and N t to the curve at the given point r t ln sin t i j tk t 6

1 Match each vector equation with its graph i r t 2 i tj tk iii r t ti t j ek U ii r t ti tj sin at k iv r t cos 2t i tj sin 3t k

se Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint If an answer does not exist enter DNE f x y z x y z x4 y 24 7 maximum minimum

The amount of trash in a county landfill is modeled by the function T x 180x 32 000 where x is the number of years since 1996 and 7 x is measured in thousands of tons a Sketch a graph of T y 40000H 38000 36000 34000 32000 y 40000 38000 36000 5 10 15 20 25 30 35 40000 38000 36000 34000 32000 y 40000 38000 36000 y 5 10 15 20 25 30 3

tomework Decide whether the given is an ic 0 3 x 3 0 5 x 3 0 2x equation 0 6

Radian 0 26 KIN SE X Degree EN 5 2

Find the angle 9 where 0 0 satisfying tan 9 3 A 5 6 B 27 3 C TV3 D T 6

1 2 Terminal Points A point P x y is given a Show that P is on the unit circle b Suppose that P is the terminal point determined by t Find sin t cos t and tan t 3 1 P 1 1 2 2 3 t 3 6 Reference Number and Terminal Point A real number t is given a Find the reference number for 1 b Find the terminal point P x y on the unit circle determined by t c Find the six trigonometric functions of t 2 3 11T 4 J5J bol 2 P 5 t 5 4 t 0 6 t 5 T 3 7T 6

13 a tan 5T 2 lopouls corsi b cot 5 T 2

4 1kg 10 9 8 7 6 5 4 3 2 1 0 1 2 T 4 2kg 3 4 5 6 4kg 7 As shown above there are 3 masses located in a line at the positions shown in meters At what position would the center of mass be located for this system of 3 masses 8 9 10

Let C be the CCW unit circle centered at the origin In each case do the following i Find the domain of F ii Calculate F T ds by hand C iii Calculate f F Tds using Green s Theorem C iv Compare your answers to Parts ii and iii Use Part i to explain the difference if applicable a F y x I F b F

a Graph the sine and cosine functions b What are the amplitude period and horizontal shift for the sine curve y a sin k x b and for the cosine curve y a co s k x b c Find the amplitude period and horizontal shift of y 3 sin 2x sin 2 TT

11 a What is simple harmonic motion b What is damped harmonic motion c Give real world examples of harmonic motion

49 52 Evaluating Expressions Involving Inverse Trigonometric Functions Find the exact value of each expression if it is defined 49 sin 1 50 cos 51 sin sin 13T 6 11141 52 tan cos 2 COS RET

18 tant sec t cos t 19 tan t sin t t in Quadrant IV

5 a What is an even function and what is an odd function b Which trigonometric functions are even Which are odd c If sin t 0 4 find sin t d If cos s 0 7 find cos s

29 36 Horizontal Shifts A trigonometric function is given a Find the amplitude period and horizontal shift of the func tion b Sketch the graph 29 y 10 cos x 31 y sin x 33 y 3 sin 2x 2 35 y cos 6 30 y 4 sin 27x 32 y TT m x 7 4 2 sin x TT 34 y cos 2 x 2 cos x 36 y 10 sin 2x

cot Start by drawing two consecutive asymptotes Between those asymptotes plot three points a point where the graph intersects the x axis point to its left and a point to its right Finally click on the graph a function button X

A pole leans away from the sun at an angle of 7 to the vertical as shown in Figure 27 When the elevation of the sun is 55 the pole casts a shadow 42 feet long on the level ground How long is the pole Round the answer to the nearest tenth anlad 7 e alt banworth 0268 A 1 I 55 42 ft Figure 27 oals at 8 st 102 2 Anal B

A projectile is fired from a cliff 190 feet above the water at an inclinati 32x h x x 190 45 where x is the horizontal distance of the projectile from the face of the

3 10 Evaluating Inverse Trigonometric Functions Find the exact value of each expression if it is defined 3 a sin 1 b sin V3 2 c sin 2

make sure im right if you could explain the steps so i can see if i did anything wrong Thank you 30 much Write an equation for the function graphed below J 7 y 6 5 4 5 Show Transcribed Text G Write an equation for the function graphed below The y intercept is at 0 0 3

Question 6 A person leaves home and walks 4 miles west then 6 miles southwest How far from home is she miles In what direction must she walk to head directly home degrees North of East

31 sin sin 7 3 4 33 sin sin 35 cos 37 cos COS 0 COS 5 T 6 7T 6

Score 20 100 1 12 answered Question 3 4 miles Southeast 3 miles South A person starts walking from home and walks 3 miles East 2 miles Southwest 3 miles East This person has walked a total of Find the total displacement vector for this walk If this person walked straight home they d have to walk Question Help Video miles miles

Write the vector shown above in component form Vector 0 Note In the graph each box is 1 unit by 1 unit in size Question Help Video Submit Question

Answer the questions below about the quadratic function g x 2x 16x 29 Does the function have a minimum or maximum value Jadratic Minimum x Maximum Where does the minimum or maximum value occur 0 What is the function s minimum or maximum value 0 00 X

pt tails A maintenance worker fills a swimming pool with water m at a rate of 4 minute Suppose at t 0 the swimming pool is filled with 70 m of water How much water will be in the swimming pool at t 12 minutes A function giving the volume V of water in the swimming pool after t minutes is V t m When will the swimming pool contain 190 m of water minutes

3 Graph the function defined by the equation f x x x Find the intercepts vertex and tell whether the vertex is a maximum or minimum point Indicate the domain and range y X

Find the equation of the following plane The plane passing through the origin and containing the line x t 1 y 2t z 3t 4

The solution to this initial value problem dy dx is U T 4e 3 cos x y 0 3

I point Suppose x y z 1 2 2 P 3 3 2 Find the gradient of f af i j Hote Your answers should be expressions of x y and z e g 3x 4y 5 What is the maximum rate of change of f at the point P lote Your answer should be a number

Find both the vector equation and the parametric equations of the line through 0 0 6 in the direction of the vector v 2 6 0 where t 0 corresponds to the given point The vector equation is x y z

tooning and PTOOT st K Complete the following proof by filling in t Given 126 Prove Z3 Z4 Statements Question 4 GA 1 41 26 2 Z4 21 3 4 23 46 5 23 24 Reasons 1 Given 2 3 Transitive property o 4 5

of Question 3 Part 2 o Use an algebraic equation to find th degree measure of the angle s supp The measure of the angle is tw What is the measure of the suppler

Which is the inverse matrix of 2 13 3 15 2 121 3 1513 5 1 2 25 1 3

Translate the sentence into a mathematical equation The total revenue derived from selling x dishwashers is 190 per dishwasher times the number of dishwashers sold Let R represent the revenue and x the number of dishwashers sold R

2 Shetch the region and find its areo a y x y 2x b y e y 2e 1 x 0

All edges of a cube are expanding at a rate of 7 centimeters per second a How fast is the volume changing in cm sec when each edge is 4 centimeter s cm sec b How fast is the volume changing in cm sec when each edge is 15 centimeters cm sec

Find the derivative of the trigonometric function y cos 5x y x 5 sin 5x X