Differentiation Questions and Answers

Suppose that a population of bacteria triples every hour and starts with 500 bacteria Find an expression for the number n t Use it to estimate the rate of growth of the bacterial population at 1 5 hours Round your answer to the nearest whole number bacteria hr n 1 5

Graph the following rational function by hand Give the domain and range and discuss symmetry and any asymptotes f x 10x Choose the correct graph below OA OB The domain is Type your answer in interval notation OC G D Q 5

if x y

Differentiate the function y y In ex xe x

4 Find f 4 if f x 5 6x f 4 Simplify your answer Type an integer or a fraction

In 4 5x find f 0 e 1 25 Enter a number

a fog b gof a y ATL Simplify your answer 2 b gof x x 2 Simplify your answer The domain of fog is 4 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression Sir your answer c fof c fof x x 124 42 Simplify your answer d gog The domain of gofis 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression Sim your answer d gog x Simplify your answer The domain off of is 0 00 Type your answer in interval notation Use integers or fractions for any numbers in the expression Simp your answer

Differentiate the function G y 2y 1 G y In 2 y 1 2y 1 4y y 6 2 2 y 1 X

Find y and y y In sec 2x tan 2x y 2 sec 2x X y 4 sec 2x tan 2x X

Find the dimensions of the rectangle with perimeter 1360 centimeters and maximum area length width area centimeters centimeters What is the maximum area square centimeters

5 The area A of a rectangle is represented by the formula A lw where I is the length and w is the width The length of the rectangle is 5 Write an equation that makes it easy to find the width of the rectangle if we know the area and the length

Find equations of the tangent lines to the curve y at the point 1 0 nt at the pointe y 7 In x X Illustrate by graphing the curve and its tangent lines e at the points 1 0 ande

Find y and y y y y In x 8 x

Perform the subtraction and find the difference 2x 0x 12x 2x 8x 6x Ax Bx A type your answer B type your answer

Let f x x x Find f 4

Problem 4 Let f x tan x Find f 3

Problem 5 Let f x x x Find f 4

Problem 2 If f x x x x 5 and f 1 2 find 1

Let I S e e 2x 1 Then for an arbitrary constant c the value of J I equals a log 2 e4x x 1 2x b log 1 c log d log ef e4x e x 1 e x e x e 1 et 1 4x dx J f les e x et 1 e x e 1 2x 4x e le4x x 1 e x 1 c C C C e 4x dx

3 Let f t cos t sin t For each a 4 define Ra to be the region between the graph of f and the t axis and between the vertical lines t 4 and t a Here we label the horizontal axis as the t axis a In the diagram below sketch the region Ra Without using the fundamental theorem of calculus determine whether the rate of change of the signed area of the region is positive for this value of a Explain your answer in terms of how the signed area is changing at this point N b Find the maximum and minimum values of the rate of change of the signed area of Ra for a 4 1 5 31

The amount of carbon 14 present in animal bones after t years is given by P t Poe 0 00012097t A bone has lost 36 of its carbon 14 How old is the bone To the nearest year what is the age of the bone Round to the nearest integer as needed

Suppose that f x 5e ex Find f 3 f 3

Find the equation of the tangent line to the curve y 9xe at the point 0 0 The equation of this tangent line can be written in the form y mx b Find m and b m b

The following table gives the values for functions f and g and their derivatives for integer values of x between 1 and 5 X 1 2 3 4 4 f x 4 2 f x 5 3 g x 5 4 2 1 g x 3 2 4 3 Let ind p 2 q 3 r 4 p x f x g x q x f x g x st 5 1 1 5 4 1 1 4 r x xf x g x X

if f 5 7 g 5 2 f 5 8 and g 5 5 find the values of the following a fg 5 b f g 5 c g f 5

Let f x 8 x 2 Find f x

Differentiate Tal P a 9

Suppose that and Find h 3 h 3 f 3 10 h x g 3 6 f x g x f 3 1 and g 3 7

By applying the Product Rule twice one can prove that if f g and h are differentiable then fgh f gh fgh fgh Now in the above result letting f g h yields Use this last formula to differentiate y e d x 3 x x

Compute the derivatives of the following functions a The Michaelis Menten kinetics function dv do b The Hill function V y Kx k x Ax an xn

Compute the derivative of the given function 12x 1 f x 4x 8x 8 4x 8x

Let f x f x Find f 4 f 4 x 7 Find f x E E

Differentiate the following function R x R x 6 28

6x x Find the derivative of F x Note that you can do this two ways either by simplifying first or by using the Quotient Rule Both approaches should obtain the same answer F x

Differentiate g x g x 1x 2x

r Let f x 4x 3 Evaluate f x at the following points A f 3 B f 8

Differentiate Y u Y u u u u5 3u

if f x 5x8 4x5 6x 6x find f x and f 1 f x f 1

Find y and y y y y x In 6x 9 6 X 2xIn 6x x 1 2In 6x 2

Find the remainder when f x x88 5x 2 is divided by x 1

0 Which of the following numbers is not a potential zero of f x 18x 9x 5x 2 3 1 e 2 2 a 2 b 2 9 c 1 1 1 or to ante

28 F v a v 1 O

Current Attempt in Given the demand curve p 132 q and the supply curve p 4 q2 find the producer surplus when the market is in equilibrium Round your answer to 3 decimal places The producer surplus is i when the market is in equilibrium

A desalination process reduces the amount of salt in a container of water by 4 per hour If 35 grams is the initial amount o alt then the amount of salt remaining after r hours is modeled by S 35 x 0 96 grams Make a graph of the amount of salt remaining after 7 hours Suggestion You may zoom in to the graph to manage points easier Double click any point to view the coordinate the point i placed Round coordinates to one decimal place where needed

A sofa regularly sells for 510 The sale price is 397 80 Find the percent decrease of the sale price from the regular price

Consumers millions y 56 54 52 Potato Chips Consumption 5 55 35 50 48 46 44 42 0 42 97 40 0 1 2 3 4 5 6 7 8 9X Years Since 2011 FOOD The graph shows the number of people in the U S who consumed between 8 and 11 bags of potato chips in a year since 2011 a Use the graph to estimate the average rate of change of consumption from 2011 to 2016 Then check your results algebraically b Interpret your results in the context of the situation

1 3 y y4 12 R t t e 3 t II F y 13 y X 1 x x y 5y 14 y X 1 x x

4 g x x ex

10 Y u u u u 2u

16 y 18 y t 1 1 S kes