Differentiation Questions and Answers

The table shows the position of a cyclist a Find the average velocity for each time period i 1 3 ii 2 3 mathrm m mathrm s mathrm m mathrm s iii 3 5 mathrm m mathrm s iv 3 4 mathrm m mathrm s b Estimate the instantaneous velocity when t 3

5 Center at 1 3 with radius 3 x 1 y 3 9 3 D x 1 y 3 B x 1 y 3 3 E x 1 3 9 C x 1 y 3

The sine and cosine curves ya sin kx and y a cos kx k 0 have amplitude period and period The sine curve y 2 sin 2x has amplitude

a c Evaluate t 1 ln t dt 1 72 2 1 In t C b t t 4 t 1 ln t t t C d 12 t C t In t 1 t 1 In t t t C

7 Evaluate In x dx c x lnx 2xlnx 2x C x lnx 2 2xlnx 2x C b d x lnx 2x ln x 2x ln x 2x C

a c Evaluate x sin xdx xcosx 2xsinx 2cosx C x cos x 2xsin x 2 cos x C b d xcosx 2xsinx 2cosx C x cos x 2xsin x 2 cos x C

y 5 cos x 1 amplitude period horizontal shift Graph one complete period y H O y 4 2 2 Vr V 3x 2x 2 2 2 4 KIN 2 4 y 4 2 27 X X y 2 7 3x

The terminal point P x y determined by a real number t is given Find sin t cos t and tan t 24 5 2 sin t cos t tan t 11

12 cos 7 T 9 COS 2 T 9 sin 7 TT 9 sin 2TT 9

ind the limit Use l Hospital s Rule if appropriate If there is a more elementary method consider using it lim X 0 e7x 1 7x x

he half life of cesium 137 is 30 years Suppose we have a 70 mg sample a Find the mass that remains after t years y t 70e In 2 xt 30 mg

A Cepheid star is a type of variable star which means its brightness is not constant The relationship between the brightness of a Cepheid star and its period or length of its pulse is given by M 2 78 log P 1 35 where M is the absolute magnitude or brightness of the star and P is the number of days required for the star to complete one cycle What is the absolute magnitude of a star that has a period of 45 days Use a calculator Round your answer to the nearest hundredth 5 95 4 60 3 25 4 60

Find the derivative h x h x In cosh 6x

ove the identity 1 tanh x e2x 1 tanh x 1 tanh x 1 tanh x 1 cosh x cosh x cosh x e x cosh x 0 ex e x ex e x

Find the derivative f x f x tanh 1 e2x

Find the numerical value of each expression Round your answers to five decimal place a sinh 0 b cosh 0

Geology The Richter magnitude M of an earthquake is related to the energy released in ergs E shown by the formula M 2 3 log E 10 11 1 How many times as much energy was released by the 1811 New Madrid Missouri earthquake than by the Fort Tejon California earthquake

11 The volume of blood plasma in adults is 3 1 L Its density is 1 03 g cm Approximately how many pounds of blood plasma are there in your body 12 A 25 0 mL sample of water at 90 C has a mass of 24 12 g Calculate the density of water at this temperature

Determine if the following integrals converge or diverge Be sure to go through the proces Guess Graph Show and Conclude 1 for 1 cos x 2 x x 1 dx

Let P x y be the terminal point on the unit circle determined by t Then sin t F cos t F and tan t

Calculate the first second and third derivatives of y x x 4x Use symbolic notation and fractions where needed y x xx 4x 5 y x 16x54x 40x44x 20x 4x y x

If f x 2x 5x 8 find f 2 f 2 Simplify your answer

If y 7x 4 x5 The value of find dy dx dy dx at x 1 at x 1 is Simplify your answ

a Find the acceleration at time t 3 min of a helicopter whose height is s t 500t 3t m Give your answer as a whole or exact number a 3 10 b Assume that a helicopter height is s t 600t 3f m Consider the graph 20 30 40 50 60 70 50 419 Incorrect 90 Does this graph show that the helicopter is slowing down for 0 6 Yes No I

mework 1 3 1 4The Derivative of e 1 13 0 16 answered Question 2 se the limit definition of the derivative to find the slop 3 valuate each of the following and enter your answers in Step 1 f 3 h Step 2 f 3 h f 3 f 3 h f 3

Steven buys a movie ticket for 14 75 and then buys some snacks for the movie He spends a total of 19 23 About how much did Steven spend on snacks Look at how you could show your work using a numberline 02 14 14815 Solution 16 02 20 APPLY IT Parker mixes 3 ounces of blue paint with 13 ounces of yellow paint to make green paint for the leaves of a tree How many ounces of green paint did Parker make Estimate and then compute Frolain how

9 14 Points on the Unit Circle Find the missing coordinate of P using the fact that P lies on the unit circle in the given quadrant Coordinates 9 P 3 1 Quadrant III

Find the value a 0 such that the tangent line to the graph of f x x e 4x at x a passes through the origin Use symbolic notation and fractions where needed a

alad Attempt 1 ion 2 0 5 points Listen cars start moving from the same point One t er travels west at 40 mph At what rate is the d reasing two hours later Round the result to the 55 42 mph 81 38 mph

A rectangle s length L t and width W 1 measured in inches are varying in time t in minutes Let A t be the area of the rectangle Determine A t at t 6 with L 6 9 W 6 4 L 6 8 and W 6 3 Give your answer as a whole number A 1 Is the area increasing or decreasing at that time O The area is decreasing at 1 6 since A 6 is positive The area is increasing at t 6 since A 6 is positive The area is increasing at 1 6 since A 6 is negative The area is decreasing at t 6 since A 6 is negative inches minute

Use the information about functions and their derivatives at x 8 to find the derivative of their product and their quotient 8 f 8 6 g 8 g 8 8 Use symbolic notation and fractions where needed g

Calculate the first second and third derivatives of y x Use symbolic notation and fractions where needed y x 2 39 x y x

Calculate the first second and third derivatives of y t 7t 8t 6 Use symbolic notation and fractions where needed y t y t

Use the table of values to calculate F 5 where F x x f x f 5 f 5 6 5 Give an exact answer Use symbolic notation and fractions where needed

Calculate the derivative 15t 1 1 Give an exact answer Use symbolic notation and fractions where needed f 1 f t

The tangent lines to the graph of f x 15x grow steeper as x increases At what rate do the slopes of the tange lines increase Give your answer as a whole or exact number ds dx 11

Evaluate the indicated limit Enter INF for INF for 00 or DNE if the limit does not exist but is neither co nor x 2 97 31 2 1 82 lim DNE help numbers

Determine whether 31 is a solution of the equation 5x 5 3x 56 Is 31 a solution OA No 31 is not a solution because substituting the value into the equation results in a true statement OB Yes 31 is a solution because substituting the value into the equation results in a faise statement OC No 31 is not a solution because substituting the value into the equation results in a false statemer O D Yes 31 is a solution because substituting the value into the equation results in a true statement

dx far x Jay Sino Sa k Determine the eighth derivative of Sxth

Find values of k where solutions are real for 19 2x x 1 2k 0 Solve the following equations 21 x x 3x 0 20 kx 16x 2 0 22 x x 6x 0

The average annual rainfall for a group of cities is 10 46 inches and the standard deviation is 1 2 inches One of these cities has an annual rainfall of 8 72 inches What is the standardized version z score of this value O 1 45 O 1 45 O 1 20 O 0 54

Which measure of central tendency is most affected by outliers O Range Median O Mean O Mode

How is the range for a set of data calculated The values are added up and the sum is divided by the sample size The lowest value is added to the highest value It is not calculated because it is the middle number O The lowest value is subtracted from the highest value

The total monthly rainfall for a city was recorded for 10 months and this data is provided below 6 7 8 5 10 3 10 4 12 6 12 9 13 1 15 5 17 8 17 9 What is the mean of this data O 11 2 O 12 9 17 9 O 12 57

Which measure of central tendency is the mathematical average of the data O Mean O Mode O Median O Range

quations K Evaluate th x x When x

If you are given the graph of h x logex how could you graph m x log x 3 O Translate each point of the graph of h x 3 units up O Translate each point of the graph of h x 3 units down O Translate each point of the graph of h x 3 units right O Translate each point of the graph of h x 3 units left

What is the true solution to the equation below In enx In ex 2 In 8 Ox 2 X 4 Ox 8 Ox 84 0000

For what value of n does 0 0 0 12 3 10 13 9 216 2 36 3n 216

What is the true solution to the logarithmic equation below log 6x log x 2 O x 0 O x 9 OX 2 0 TO 0 x 3 X A