Application of derivatives Questions and Answers

After an electric power interruption the current i in a circuit is given by the equation shown below where t is the time Find the expression for the total electric charge q to pass a point in the circuit if q 0 for t 0 i 21 1 e e

Integrate the function Sco 5 cos x sin x dx 5 cos x sin x dx

Evaluate the following definite integral T 36 T 36 0 cos 6x CSC 6x cos 6x dx dx

Integrate the function 13 tan xsec xdx

Evaluate the given definite integral 2x 6 x dx 3 3 1 3 x 6 x dx 3 Type an integer or a simplified fractic

Find the area under the curve y 5x that is between the lines x 1 and x 4 The area under the curve y 5x that is between the lines x 1 and x 4 is Simplify your answer

Evaluate the following indefinite integral S 8x 7 8x 7 7 X 7 7 X dx dx

Exam Type Question Big Bang Dimensional Analysis When Sheldon s barber becomes ill he complains about his hair growing at an astonishing rate of 4 6 yoctometers ym per femtosecond fs What is this in inches month Note 1 month 30 days 1 ym 10 24 m 1 in 2 54 cm exact 4

to see the correlation between test scores and homework The homework grade x and test grade y are given in the accompanying table Write the linear regression equation that represents this set of data rounding all coefficients to he nearest tenth Using this equation estimate the homework grade to the nearest integer for a student with a test grade of 38 Answer Attempt 1 out of 2 Homework Grade x Test Grade y 46 55 50 59 76 87 87 90 Copy Values for Calculator Open Statistics Calculator 55 46 74 90 80 89 12520

I Y 2 1 5 Select Select Not a function Function O 2 3 2 1 5 7

Determine the amplitude of the following graph 27 5 4 3 2 1 2 3 4 5 T 27 5 37 47

Which equation represents a line which is perpendicular to the line y 2 Answer 5x 4y 28 4 5 12 4x5y 25 4x 5y 30

What is an equation of the line that passes through the point 4 4 and is parallel to the line 3x 4y 16

ind the distance between the two points in simplest radical form k 10 9 8 7 6 5 O T O 3 2 1 10 9 8 T 7 10 10 3 6 2 33 4 5 5 8 N 5 6 2 3 4 5 6 7 8 9 10

Write the quadratic equation in standard form 22 14 7r

Graph the inverse of the provided graph on the accompanying set of axes You must plot at least 5 points Click the graph to make a point Click it again to erase 10 9 8 7 6 5 4 3 2 y 10 1 9 8 7 6 5 4 3 2 1 1 2 3 5 8 10 1 2 3 4 5 6 7 8 9 10 X

Evaluate the function graphically O 10 9 8 7 6 5 4 3 2 1 10 9 8 7 6 5 4 2 1 1 2 3 4 y 56 7 8 9 10 1 2 3 4 5 6 7 8 9 10

Under her cell phone plan Hawa pays a flat cost of 51 50 per month and 3 per gigabyte She wants to keep her bill at 64 10 per month Write and solve an equation which can be used to determine g the number of gigabytes of data Hawa can use while staying within her budget Answer Attempt 1 out of 2 Equation B

z x y e ind dz dt using the chain rule Assume the variables are restricted to domains on which the functions are defined dz dt z 4t y 4 5

1 point Suppose z x sin y z 18 3t y 2st A Use the chain rule to find and as functions of x y s and t z 8s z Ot as B Find the numerical values of dz s 3 1 3 1 and when s t 3 1 dt

Find an equation of the tangent plane to the surface z 1x 3y 3x 3y 3 at the point 3 5 87 22 11

Estimate f 2 01 1 97 given that f 2 2 2 fr 2 2 3 and fy 2 2 1 f 2 01 1 97

Derive the formula for the volume of a right circular cone of height h and radius r using an appropriate solid of revolution The volume of the solid generated by revolving the region between the x axis and the graph of a continuous function y f x 20 Lsasx b radius height about a vertical line x Lis V 2 2x the volume of a right circular cone shell shell dx This is known as the shell method How can the shell method be used to fin O A Rotate a shell in the shape of a quarter circle with radius r about the y axis OB Rotate a shell in the shape of a triangle with height h and base radius r about the y axis OC Rotate a shell in the shape of a rectangle with height h and base b about the y axis

6 To reach Platinum Medallion status with Delta Airlines you need to fly at least 75 000 miles but less than 125 000 miles in a given year Express the set of annual miles you could earn x to get Platinum status in algebraic interval and set notation

YA 9 8 7 6 5 1 1 y f x MA Nigh y g x 2 3 4 5 6 7 89 EXAMPLE 4 Using graphs to evaluate composite functions Use the graphs of fa g in Figure 1 9 to find the following values a f g 3 b g f 3 c f f 4 d f g f 8 e g g 9 f g f g 6 X

7 10 points The following figure shows the level curves of z two variables P where fr is positive Q where fr is negative 0 R where fy is zero S where fy is negative 10 20 f x y for f x y a function of draw and label points with the following properties If it is not possible to draw such a point don t do so 30 40

21 32 Find the derivative of the function using the definition of derivative State the domain of the function and the domain of its derivative 21 f x 3x 8 22 f x mx b

3 A rechargeable battery is plugged into a charger The graph shows C 1 the percentage of full capacity that the battery reaches as a function of time elapsed in hours a What is the meaning of the derivative C 1 b Sketch the graph of C 1 What does the graph tell you Percentage of Jull charge 100 80 60 40 20 t hours

1 2 Use the given graph to estimate the value of each derivative Then sketch the graph of f 1 a f 0 e f 4 b f 1 f f 5 ya 1 0 1 y f x c f 2 g 6 X d f 3 h f 7

f x is continuous for 0 5 x 0 2 and also has the following values X f x 0 5 0 4 0 3 0 2 1 1 1 1 3 1 6 Which of the following may be true about f x I f x may be monotonic for 0 5 x 0 2 II f x may be increasing for 0 5 0 2 III f x may be concave up on 0 5 0 2 A I only B II only C III only D I and III only E I II and III

3 Match the graph of each function in a d with the graph of its derivative in I IV Give reasons for your choices b YA VA I III YA 4 4 4 d II IV 3 YA 0 x

3 The displacement in meters of a particle moving in a straight line is given by the equation of motion s 1 1 where t is measured in seconds Find the velocity of the particle at times t a t 1 t 2 and t 3

D a Find the slope of the tangent to the curve y 2 x at the point where x a b Find equations of the tangent lines at the points 1 2 and 9 6 c Graph the curve and both tangents on a common screen

If the function f has a continuous derivative on 0 c then ff x dx A f c f 0 B f c f 0 C f c D I f c f 0 I E f x C

Let f x x 9 X 6 g x X X 6 Express f x g x as a rational function f x g x for x Simplify your answers Use a comma to separate answers as needed

Describe the domain of the function g x 1 6 x The domain of the function consists of all real numbers x where Type an inequality

Find an equation of the line such that the points An equation of the line containing the points Type an equation 5 and 3 4 4 are on the line 3 5 and 4 is 5

6 If f x X find f x h f x and simplify f x h f x

If f x x 5x find f 0 f 4 f 5 and f 8 f 0 Simplify your answer f 4 Simplify your answer f 5 Simplify your answer f 8 Simplify your answer

Let f x x 8 h x 4 4x Calculate the following function f x h x f x h x

The quadratic formula is used to solve for x in equations taking the form of a quadratic equation ax bx c 0 b b 4ac 2a Solve for x in the following expression using the quadratic formula 2x 35x 5 1 0 Use at least three significant figures in each answer x quadratic formula x I and x

2 a Can the graph of y f x intersect a vertical asymptote Can it intersect a horizontal asymptote Illustrate by sketching graphs b How many horizontal asymptotes can the graph of y f x have Sketch graphs to illustrate the possibilities

What is the minimum value of f x 3xe OA 3e OB 338 e O C 1 D 0 E f x has no minimum value

Given that f 0 5 2 and f 0 5 4 using a tangent line approximation you would estimate f 0 to be A 0 B 3 OC 2 D 1 E 4 E

X y The sides of the rectangle above increase in such a way 1 and da d 3d At the instant when x 4and y 3 what is the value of dt dt A 5 B 5 C 2 Z D 3 E 1

Consider the function x whose graph is composed of two intersecting line segments shown below Which of the following are true for f x on the open domain a c I f x is continuous on a c II f x is continuous on a c III f x is differentiable on a b A I II and III only B I and III only C III only D II only E I only

Water is pouring into a container at a rate given by ri t t gallons hour water is pouring out of the same container at a rate given by ro t et gallons hour OA 41 608 B 10 402 OC 89 598 If there are 100 gallons of water at time t 0 how many gallons of water will be in the container at t 4hours D 110 402 2 OE 141 608 and

An equation of the line tangent to the graph of f x x 1 2x at the point 1 1 is A y 6x 5 B y 2x 1 C y 2x 3 D y 7x 6 E y 7x 8

If the function f is continuous for all real numbers and if f x x 2 then f 2 A 0 B 2 C 1 D 4 E 2 x 4 x 2 when

2 In this problem you will investigate the family of functions h x x cos ax where a is a positive constant such that 0 a 4 A Graph the curves y x x cos x and y x x cos 4x in the space provided below 4 B For what values of a will h x have a relative maximum at x 1 C For what value s of a will h x have an inflection point at x 1 I