Calculus Questions

8 For the function A whose graph is shown state the following a lim A x b lim A x x x 3 c lim A x x 2 e The equations of the vertical asymptotes 3 d lim A x x 1 YA 0 I 2 5 18

7 For the function g whose graph is shown find a number a that satisfies the given description a lim g x does not exist but g a is defined x a b lim g x exists but g a is not defined x a c lim g x and lim g x both exist but lim g x does x a ca xa not exist d lim g x g a but lim g x g a x a x a YA 0 2 7 I 4 6x

3 Let F x 3r cos 1 2 dt A Using the First Fundamental Theorem of Calculus find F x Yes this is as easy as it seems B Starting with F x f 31 cos 1 2 dr use the substitution u 1 to rewrite the definite integral You should get a new equivalent expression for F x which consists of this new definite integral

4 Use the given graph of f to state the value of each quantity if it exists If it does not exist explain why a lim f x x 27 d f 2 b lim f x x 2 4 e lim f x x 4 YA 2 0 2 4 48 c lim f x x 2 f f 4

4 The function has domain below IX B Find h y f 1 5x 2 which it is continuous A graph of y is given Define a new function h x by h x f t dt A What is the domain of x that is on which x values is the function x defined

beginning of a race Velocities of the two cars during the race are given in the tables below where velocities are in miles per hour and times are in seconds Sappho s car timer velocity vs 0 1 2 4 5 6 7 8 9 10 0 20 35 48 85 93 99 106 Homer s Car time 0 1 2 4 6 7 8 10 velocity VII 0 18 43 68 79 86 93 95 96 A Use ten trapezoids to approximate the area between the curves y vs t and y vu 1 on the interval from 0 to 1 10 What definite integral are you approximating B If you haven t done so convert your answer from part A into units that make good common sense for the physical situation described Then explain what practical meaning this answer has for the physical situation

1 Explain in your own words what is meant by the equation lim f x 5 x 2 Is it possible for this statement to be true and yet f 2 3 Explain

If a rock is thrown upward on the planet Mars with a velocity of 10 m s its height in meters t seconds later is given by y 10t 1 86t 1 a Find the average velocity over the given time intervals ii 1 1 5 iii 1 1 1 i 1 2 iv 1 1 01 v 1 1 001 b Estimate the instantaneous velocity when t 1

Consider the piecewise defined function that is graphed above The point A 3 2 0 a Find the equation of the piecewise defined function graphed above You may assume that the ends of the pieces shown end with an open circle an empty point b Calculate the inverse of each of the pieces by swapping z and solving for y and

12 Let the function u t be defined by u t at b ct d with arbitrary real valued constant coefficients a b c d a Determine the domain for generic values of a b c d Are there any special value of these coefficients for which the domain is larger b Determine a condition on the coefficients such that u is injective and find th inverse function u function in that case Hence conclude what is the range of the function u for generic coefficients a b c d

5 The deck of a bridge is suspended 275 feet above a river If a pebble falls off the side of the bridge the height in feet of the pebble above the water surface after t seconds is given by y 275 16t a Find the average velocity of the pebble for the time period beginning when t 4 and lasting i 0 1 seconds ii 0 05 seconds iii 0 01 seconds b Estimate the instantaneous velocity of the pebble after 4 seconds

The point P 2 1 lies on the curve y 1 1 x a If Q is the point x 1 1 x find the slope of the secant line PQ correct to six decimal places for the following values of r ii 1 9 vi 2 1 b Using the results of part a guess the value of the slope of the tangent line to the curve at P 2 1 i 1 5 v 2 5 iii 1 99 vii 2 01 iv 1 999 viii 2 001 c Using the slope from part b find an equation of the tangent line to the curve at P 2 1

A student bought a smartwatch that tracks the number of steps she walks throughout the day The table shows the num ber of steps recorded minutes after 3 00 PM on the first day she wore the watch t min Steps 10 20 30 40 3438 4559 5622 6536 7398 0 Matala BORDE PAPINH AMITRIP Helen www ke a Find the slopes of the secant lines corresponding to the given intervals of t What do these slopes represent ii 10 20 iii 20 30 i 0 40 b Estimate the student s walking pace in steps per minute at 3 20 PM by averaging the slopes of two secant lines

Write the slope intercept equation for the line having the given information 1 3 and 2 3 on the line What is the slope intercept equation of the line Use integers or fractions for any numbers in the equation

3 Use the power rule to find the derivative of f x x fi

Estimate the price of one barrel of crude oil on January 25 2016 and the rate at which the price was rising on that day Your answer for the rate should be in dollars per day The price of one barrel of crude oil on January 25 2016 was about Type an integer or a decimal rounded to two decimal places as needed Barrel Price of Crude Oil 30 29 28 27 26 25 24 23 0 5 10 15 20 25 30 January 2016 o o

pi 1 Find the derivative of f x 3 10 at x 3 10 Type an integer or a simplified fraction

Find the slope of the graph of y f x at the designated point f x 2x 6x 2 3 2 The slope of the graph of y f x at 3 2 is

Use limits to compute the derivative 2 f 8 where f x x 7 f 8 Simplify your answer

If f x 1 10 X compute f 2 and f 2 f 2 Simplify your answer

Use limits to compute the derivative 3 f 0 where f x x 6x 2 0 Simplify your answer

Differentiate d dx y 3x 8 5 3x 85

Differentiate d dx 6 f x x x x 4 x x x

Find the equation of the tangent line to the graph y f x at the given value of x f x 10x 8 x 6 What is the equation of the tangent line Type an equation CO

Find an equation of the tangent line to the graph of y f x at the given x 1 49 f x x x What is the equation of the tangent line Type your answer in point slope form

Use the power rule to find the derivative of f x x6

Use the power rule to find the derivative of f x 2 x f x

Find the equation and sketch the graph of the following line With slope 2 and y intercept 0 5 www

Find an equation of the line below Slope is 4 8 4 on line

What does the transformation fly 26 da

What does the transformation f x f x do to the graph of f x reflects it across the y axis shrinks it vertically stretches it vertically reflects it across the x axis

7 1 point Sally is running a sunblock factory and there is a machine to fill the sunblock bottles The machines can fill 150 bottles an hour and each bottle takes 20 ounces of sunblock If Sally runs the machines for 10 hours at least how many gallons will Sally need to fill all of the bottles 204 gallons 234 gallons 235 gallons 213 gallons Previous Next

1 Consider the region bounded by the curves y x x 12 x 5 and x 5 and the x axis A Set up a sum of integrals not containing an absolute value symbol that can be used to find the area of this region I B Find the area of the region by using your answer from part A Don t approximate with

4 Consider the curves y x and y mx where m is some positive constant No matter what positive constant m is the two curves enclose a region in the first quadrant Without using a calculator find the positive constant m such that the area of the region bounded by the curves y x and y mx is equal to 8

3 Consider the function f x x sin x on the interval 0 which is graphed below 0 8 0 6 0 4 0 2 0 4 0 6 0 8 I y xsin x A Find a constant function g x constant on the interval 0 such that the area under the graph of g is the same as the area under the graph of fon the interval 0 Hint Remember that a constant function g x c is just a horizontal line

sing of a certai is given b where P is in dollars and M x is marginal cost in dollars per unit A Suppose the marginal cost at a production level of 500 units is 10 per unit and the cost of producing 500 units is 12 000 that is M 500 10 and P 500 12000 Use a tangent line approximation to estimate the cost of producing only 497 units B Suppose the production schedule is such that the company produces five units each day That is the number of units produced is x 51 where is in days and 0 corresponds the beginning of production Write an equation for the cost of production P as a function of time t

Find the area of the region IN THE FIRST QUADRANT upper right quadrant bounded by the curves y sin x cos x y 2x cos x and y 4 4x Use the area to the left of curve y 4 4x You get O A 0 16165 OB 1 7281 OC 0 36974 O D 1 8467 OE 0 37859

1 Practice finding the derivatives of functions that are defined by definite integrals 53 A Let g u 3 2 x dx Then d du B Find 3x 1 dt 20 cos 3xx

Find the area between the curve y interval 4 x 4 You get O A 184 OB 16 O c 212 OD 24 40 OE 3 2 and y x 3 on the

Find the area between the curve y 2 and y x 3 on the interval 4 x 4 You get O A 184 OB 16 O c 212 OD 24 40 OE col

The region R is the region in the first quadrant bounded by the curves y x 4x 4 2 0 and x 2 as pictured below Find a value h such that the vertical line x h divides the region Rinto two regions of equal area You get h OA 0 44123 OB 0 42739 C 0 39987 OD 0 43120 OE 0 41260

The area of the region bounded by the curves y f x 2x 6x 2x 6 and y g x x 3x x 3 is given by A OB 3a 3x x 3 da 3x 3x x 3 da 3a 9x 3x 9 da 3x 92 3x 9 da L 3x 9x 3a 9 da 32 9x 3x 9 da 32 9x x 3 da 3 3a 9a x 3 da E 3x 9x 3x 9 dx

1 Solve each of the following trig equations 0 ses 360 Draw diagrams 3 b tan 0 0 a sin 3 marks eas 2 c cos 8

The definite integral of f x from a to b is defined to be b A So f x dx limn n Wh xk f a cb B f f x dx limn k 1 wk A k f cb O c f f x dx F a F b D f f x dx limn h 1 Ax wk k 1 a b O E f f x dx F b F a

The definite integral of f x from a to b is defined to be O A f f x dx limn n f wh k O B f f x dx limn k 1 wk A k a 1 c f f x dx F a F b b D f f x dx limn 1 Ax wk k a k n ab O E f f x dx F b F a a

Factor the given polynomial x 16x 55 x 16x 55

Assume that a couple invests 750 upon the birth of their daughter Assume that the investment earns 6 4 compounded annually What will the investment be worth on the daughter s 10th birthday The investment will be worth approximately on the daughter s 10th birthday Do not round until the final answer Then round to the nearest cent as needed

Use the laws of exponents to simplify the algebraic expression Your answer should not involve parentheses or negative exponents 2x 5y 2x3 5y 3 3 Simplify your answer Use positive exponents only

For a function y g x the differential dy is given by O A dy g x dx OB dy g u dx O c dy g x dx O D dy dda O E dy da

Solve by the quadratic formula and simplify 3x x 2 0 X Use a comma to separate answers as needed