Application of derivatives Questions and Answers

Use term by term differentiation or integration to find a power series centered at x 0 for 6x5 1 x6 f x

The graph of a radical function f x is shown below y f x k a x h then the product of a h k

Assume the acceleration of the object is a t 32 feet per second per second Neglect air resistance With what initial velocity must an object be thrown upward from ground level to reach the top of a national monument 510 feet Round your answer to three decimal places ft sec

For the given functions a express 2x w 6x 3y x cost y sint t 3 dw a Express as a function of t dt dw dw 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 to t Then b evaluate at the given value dt dt dt

dz se the functions below to a express and dz du as functions of u and v both by using the Chain Rule and by expressing z directly in terms of u and v before differentiating Then b evaluate and dv z 2 tan 1 z dz a Express and as functions of x and y Find dx dz dy dz x u cos v yusin v u v 0 8 ax Type an exact answer dz du dz av at the given point C

3 You can either row your boat from point A to point C Or you can row from point A to some point P between B and C and then run the rest of the way to point C If you row at an average rate of 3 km h and run at an average rate of 5 km h what is the shortest amount of time it will take you to get to point C 15 points 5 km

How much work is done in J when a weight lifter lifts 130 kg from 1 5 m to 2 0 m above the ground Use 9 8 m s for g J

Compute the work done in ft lb in hoisting a 1 500 lb grand piano from the ground up to the third floor 30 feet above the ground ft lb

Your brother has the flu At 6 p m you checked his temperature and it was 100 1 F You then check it at 7 PM and it has increased to 101 F You know a dangerous temperature is anything above 103 a If his temperature continues to increase at this rate what time should you take him to the hospital Create a table of values to justify your conclusion it doesn t have to be pretty just make sure it s clear what s going on b Use your table to calculate the slope and write an equation in slope intercept form c What does the slope mean in this problem d What does the y intercept represent Does it have meaning in this problem e Use your equation to graph a line Mark the place on your graph time when your brother s temperature is critical What are the coordinates of this point f Should you go to the emergency room now or can you wait until tomorrow morning g Using this equation determine what time his

Among all rectangles that have a perimeter of 64 find the dimensions of the one whose area is largest Write your answers as fractions reduced to lowest terms Answer Keypad Keyboard Shortcuts

A projectile is fired with an initial speed of 220 m s and angle of elevation 60 The projectile is fired from a position 90 m above the ground Use g 9 8 m s2 Round your answers to the nearest whole number a Find the range in m of the projectile 4276 7 x m b Find the maximum height in m reached 1942 m c Find the speed in m s at impact 224 m s

Tutorial Exercise The position function of a particle is given by r t t 4t t 16t When is the speed a minimum Step 1 To find when the speed is a minimum we need to find the speed as a function of t then find its derivative and see when it v t r t 2t 4 2t 16 2t 4 2t 16 Step 2 We remember that the speed is the magnitude of the velocity vector and is calculated as follows Iv t 2t 4 2t 16 V 8 Step 3 Next we use the Chain Rule to find the derivative of the speed v t 8 64t 272 1 2 8t 32 X 2 64 t 272 272

A gun has muzzle speed 158 m s Find two angles of elevation in degrees that can be used to hit a target 800 m away Enter your answers as a comma separated list Round your answers to one decimal place Use g 9 8 m s 30 4 59 6 a X 0

Find the total area of the shaded regions 10 164 Ay f x 64 1 X 10

Use the Substitution Formula to evaluate the integral 0 1 8 7 1 0 1 1 1 18 7 dt

What is the total resistance for a circuit with three resistors in series if the resistances are 2 ohms 5 ohms and 4 ohms respectively Draw a diagram of these resistors in series What is the total resistance for a circuit with two parallel resistors one with a resistance of 3 ohms and the other with a resistance of 7 ohms 4 Draw a diagram of these resistors in parallel 5 Consider a circuit with two parallel resistors a Solve the total resistance equation for Rr Also express it as one fraction

Evaluate the integral 4e4t S4e 2 e4t 4e4t dt

Use substitution to find the given indefinite integral Sx 9x 7 dx 2

4 Find the sum given below 10 51 3 4

11 Solve the differential equation df 16u 3 du 8u 3u 2

function is given below Determine the average rate of change of the function between x 5 and x 2 h x x 2 x

Find F x given x 1 F x f 2z 5 dz

Find the equation of the line tangent to the graph of y sin x at x 0

u 7 du Describe the most appropriate substitution case and the values of u and du Select the correct choice below and fill in the answer boxes within your choi A Substitute u for the quantity in the denominator Let v so that dv du B Substitute u for the quantity under the root Let v u 7 so that dv 1 du C Substitute u for the quantity in the numerator Let v so that dv du Use the substitution to evaluate the integral

For the given functions a express as a function of t both by using the Chain Rule and by expressing w in terms dw of t and differentiating directly with respect to t Then b evaluate at the given value of t dt w 7yex Inz x In 1 y tant z e5t t 1 a b dw dt dw 14t tan dt It 1 dw dt t 2 Type an exact answer CODE

Find the total area of the shaded regions 16 Q Q The total area of the shad Type an integer or a simp

se the Substitution Formula to evaluate the integral 0 1 4 14 dt Simplify your answer 0 S 4 14 d dt 1

Use substitution to find the indefinite integral 3x x dx 8x x 8x

Evaluate the expression 14 7 8x2 9 14 7 8x2 9 Simplify your answer

1 D 1 2 5 6 3 4 Graph of f The figure above shows the graph of a function fi which has a vertical tengont at x 2 and a horizontal tangent at X 4 Which of the foroving statements is fabe fis not differentiable at x 1 because the graph of thes Jump discontinuity at x 1 fs not differentiable at X 2 because the graph of has a vertical tergent et X 2 Gf is not diffentiable at x 6 because the graph off hes I removable discontinuity e x b Of is not differentiable 24 x 4 because the graph of f has a horizonta tangent at X 4

Let F x x 81 Ix 91 Find the followu following Tim X 79 F X lim F x X 9 18 and 9 618 and 18 G both 18 limits a

use continuity to evaluate the limit Sin x 4 5in x lim X 7137 6 13T 1

Find the lim lim x 2x X 2 X 4x 4 X 2 7 2

Find the limit lim x 3 x x 6 X Select the correct ans 200 1 2

Find the given derivative by finding the first few derivatives and observing the pattern that occurs sin x d 30 dx 30

ind d 2020of where for xex dy 2020 dx

Determine where of is discontinuos if x 0 if 0 x 3 x 3 f x 2 x X 3 X 3 X if

Evaluate the integral 2 2 X 0x X 2 dx 2 x 4x 7 X 2 4x 7 dx

Draw a graph to match the description given G x is decreasing over 0 0 and 7 00 and increasing over 0 7

Use the Substitution Formula to evaluate the integral 3 413 0 4 14 dt 2

Evaluate the definite integral 9 10y y 24 y dy

Find the indefinite integral fin s In 9x 8 dx C

The rate of growth of a population in millions per year in the year x is approximated by f x xe a Find the total population function Remember that the population is given in millions b As time goes on what will happen to the total population a Set up the appropriate integral that can be used to find the population function The total population in the fourth year is 70 000

Find the given indefinite integral State whether integration by substitution or integration by parts was us f5x 9dx Using integration by Type an exact answ 5x 9 substitution dx

Use substitution to find the indefinite integral 1 1 301 151 d t e dt Describe the most appropriate substitution case and the values of u and du Select the correct choice below and fill in the answer boxes within ye OA Substitute u for the quantity under a root or raised to a power Let u so that du dt so that du OB Substitute u for the exponent on e Let u OC Substitute u for the quantity in the denominator Let u dt so that du dt

Evaluate the integral e t Se 7 t t dt

Use substitution to find the indefinite integral f8 8x 8x 11 dx

What is f x given the integral function f x cos e dur o f x 2x cos ex o f x x cos ex o f x 2x cos ex o f x x cos ex 35 36 37 38 39 40

Polar equation r 2sin 30 Outside of the unit circle r 1 See graph Find the area of the shaded part 3

Find the volume of the solid under the graph of the function over the given rectangle f x y 5 3x 2y R x y 0 x 1 0 y 1 The volume is 10 3 Type an integer or a fraction