Calculus
Differentiation
Answer Formats f x 2x 15x 84x 12 x If the function does not have any enter NONE If the function has more than one type your answer as a comma separated list For example m
Calculus
Differentiation
Oil leaked from a tank at a rate of r t liters per hour The rate decreased as time passed and values of the rate at two hour time intervals are shown in the table In this problem we are going to estimate the total amount of oil that leaked out during this ten hour time period t hours 0 2 4 6 8 10 r t L hr 9 4 7 7 7 3 5 9 4 8 4 1 Rate of Oil Leakage Left Endpoint Approximation Estimate the total amount of oil that leaked out using left endpoints Total Amount of Oil Right Endpoint Approximation Estimate the total amount of oil that leaked out using right endpoints Total Amount of Oil liters liters
Calculus
Differentiation
The limit below is a definition of f a Determine the function f x and the value of a lim h 0 f x 2 5 h 0 4 h
Calculus
Differentiation
Determine whether lim f x exists If so give the limit X 2 What is the value of the limit Select the correct choice below and if necessary fill in the answer box to complete your choice A The limit is OB The limit does not exist 3 2
Calculus
Differentiation
Calculus
Differentiation
Directions Use the equation given below to answer the following question Derivative Use implicit differentiation to calculate da dy Note that we are calculating the derivative of z with respect to y the opposite of what we normally dy This means we need to regard y as the independent variable and a as the dependent variable x y xy 2xy 0 da
Calculus
Differentiation
Directions Use the equ Part 1 Derivative dy Calculate the derivative by implicit differentiation da dy da Part 2 Evaluation Evaluate the derivative at the point 0 2 dy da 0 2 x 4xy y 4
Calculus
Differentiation
Determine the point on the graph of the unit circle that I I 2 corresponds to Then findcos andsin and state 2 which functions are undefined for I cos 2 KIN 0 2 I 2 The point on the unit circle is 0 1 Type an ordered pair Type an integer or a simplified fractio T T Simplify your answer sin 1 Simpl 2 1 0 2 cotangent tangent Ay 0 1 0 0 1 1 0 O
Calculus
Differentiation
otated through 180 Assume the radius of the bicycle wheel is 12 6 inches icycle will travel approximately in nd to the nearest tenth 1 45 in
Calculus
Differentiation
Compute the difference quotient f x h f x h f x h f x h for the function f x 3x 2x 4 Simplify your answer as much as possible
Calculus
Differentiation
Find 1 dy if y x 12 dx dy dx Type an exact answer
Calculus
Differentiation
Find 9 10 d dx d x 10 dx Type an exact answer
Calculus
Differentiation
If f x x compute f 3 and f 3 f 3
Calculus
Differentiation
Find the derivative of f x f 9 1 at x X II 7 l os 9 Type an integer or a simplified fraction
Calculus
Differentiation
Find the slope of the curve y x at x 5 The slope of the curve at x 5 is f 5
Calculus
Differentiation
1 Find the derivative of f x x8 at x 2 1 5 Simplify your answer
Calculus
Differentiation
13 Use the power rule to find the derivative of f x x
Calculus
Differentiation
Use the formula s rot to find the value of the missing variable T 4 radians per sec r 12 cm t 16 sec S cm Type an exact answer using as needed
Calculus
Differentiation
The tangent line to the graph of y x at the point x y has slope 3x Using this result find the slope of the curve at the point 3 27 The slope of the curve at the point 3 27 is
Calculus
Differentiation
Convert the following angle to degrees minutes and seconds form a 83 7022 The answer is 0 Simplify your answers Round to the nearest second as needed
Calculus
Differentiation
Write the equation of the tangent line to the graph of y x at the point x 0 9 Tyne an equation
Calculus
Differentiation
Directions Consider the following function f x at a 8 Part 1 Difference Quotient If h 0 then the difference quotient can be simplified into the form f 8 h f 8 8 h where A and B are constants Find these constants A and B Note It s possible for one or more of these constants to be 0 f x x 7 f 9 1 VA h B Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f 8 f 8 lim h 0 That is 1 A h B f 8 h f 8 h
Calculus
Differentiation
Directions Use the graph of function f x given below to estimate the following values of its derivative J x Note that a portion of the tangent line has been drawn for you at x 2 1 Estimate the value of f 4 2 Estimate the value of f 2 3 Estimate the value of f 1 ello 5 4 f x 2 3 4 5 H
Calculus
Differentiation
Calculate the derivative of the function 71 H u u V u v 1
Calculus
Differentiation
Part 1 Difference Quotient If h 0 then the difference quotient can be simplified into the form That is f x f x h f x h Note It s possible for one or more of these coefficients to be 0 1 x 4 1 x 4 A Bh 1 x 4 A Bh where the coefficients A and B can be functions of a or just constant terms Find these coefficients A B Part 2 Derivative Use the simplified expression from Part 1 to then calculate the derivative f x f x lim h 0 f x h f x h
Calculus
Differentiation
Compute the following limits Algebraically 1 1 2 x x 2 X 2 a lim 4 x h 4 x 2 c lim wallot 2 0 0 0csc0 b lim bir 1019 digs brin alogad
Calculus
Differentiation
If 2x g x x x 2 for all x evaluate lim g x X 1
Calculus
Differentiation
4 Each angle measure is given in radians Write the corresponding angle measure in degrees a 0 18 4 3 d 0 8 b 0 C 0 716
Calculus
Differentiation
Find the length and direction when defined of u xv and vxu u 21 2j 3k v 3i 3k The length of uxvis 9 Type an exact answer using radicals as needed Select the correct choice below and if necessary fill in the answer boxes to complete your choice 1 3 Type exact answers using radicals as needed The direction of u xv is 2 3 i OB The direction of u xv is undefined j 2 3 The length of vx u is Type an exact answer using radicals as needed k
Calculus
Differentiation
Find the length and direction when defined of u xv and vxu u 2i 2j 3k v 3i 3k The length of u x v is Type an exact answer using radicals as needed
Calculus
Differentiation
6 An angle in standard position has a measure of 5 8 radians In which quadrant is the terminal side of the angle
Calculus
Differentiation
1 x 0 10 3 0 1 x t 1 1 0 22 Please enter a complete math problem so we
Calculus
Differentiation
Consider the velocity function v t 4 sint for st 2 Complete the sentence The same distance could have been traveled over the given time period at a constant velocity of The same distance could have been traveled over the given time period at a constant velocity of Type an exact answer in terms of
Calculus
Differentiation
Calculus
Differentiation
Verify that the trigonometric equation is an identity cos a sec a cos acsc a csc a Which of the following statements establishes the identity OA cos a sec a cos2acsc a cos x OB OC cos a sec a cos x csc cos x 2 cos2x sec cos2a csc sin x 1 COS 1 2 COS 1 2 COS 1 sin 2 sin 1 2 sin cos 2 cos COS 2 sin x 2 1 2 a sin 1 2a sin 1 2 2 cos a sin csc 2 csc csc x
Calculus
Differentiation
12 13 15 Write each expression in radical form 16 7 12 56 A b C 5b Simplify 13 64x4 A x4 C 512x6 14 14 Tm Write each expression in exponential form 1 A 3m C 7m Simplify 3 15 x 10 5 A x5 C 10x B 106 D 2b B 1024x 5 D 6x 1 16 6b A 6b C 376 B m D m B 27x D x Write each expression in radical form B 46 D 3b 2024 Kuta Softw a re LLC All rights reserved Made
Calculus
Differentiation
Find the area of the region R bounded by the graph of f x 9 x and the x axis on the interval 9 18 Graph and show the region R Choose the correct graph below The region R is shown in yellow O A 36 36 Ay 36 X 36 OB Ay 36 36 36 Q O C 36 36 Ay 36 36 OD 36 in
Calculus
Differentiation
2 sin 2t Using a table of values the limiting value is to sin 2t 2 t cos 2t Consider lim Submi Enter DNE if the limit does not exist
Calculus
Differentiation
Tutorial Exercise Determine the infinite limit lim X 10 x 9 X 9 Step 1 As x approaches 9 from the left the numerator approaches Also as x approaches 9 from the left the denominator approaches from the Select
Calculus
Differentiation
Let cos x a Identify a Choose the correct graph below OB O A O D d O C x
Calculus
Differentiation
28 Do not evaluate
Calculus
Differentiation
Let a and b be positive numbers such that the function ax x 2 0 x 2 f x bx a Then write the value of f 8 3 x 2 is differentiable for all x 0
Calculus
Differentiation
Identify the transformations of the graph of f x that produce the graph of the given function g x Then graph g x on the same coordinate plane as the graph of f x by applying the transformations on the asymptotes x 0 and y 0 to the reference points 1 1 and 1 1 Also state the domain and range of g x using inequalities g x 2 1 2 6 5 4 3 4 5
Calculus
Differentiation
Graph the equation using the slope and the y intercept 10 y 3x 2 Use the graphing tool to graph the line Use the slope and y intercept when drawing the line Click to enlarge graph 20 16 12 8 20 16 12 2 ob 8 4 4 8 12 8 12 16 X 20 Q
Calculus
Differentiation
Use the graph of f x log 3x to graph the function f x log 3 x 2 4 Use the graphing tool to graph the function Click to enlarge graph
Calculus
Differentiation
Find the derivative f ex for each function given be a f x 1 e 1 ex x b f x Inx fex 5x3 Cosx sinx 4
Calculus
Differentiation
Given g x is a transformation of the graph f x 2x determine which of the below correctly describe g x f x 2 54 3 2 1 N 7 21 3 S Asymptote y 0 123 g x 2 x g x 2 x g x 2 x g x 2x 1 Asymptote y 0 5 4 3 2 1 21 3 g x 1 2 3 4
Calculus
Differentiation
3 Match the graph of each function in a d with the graph of 3 Match the graph of each function in a d with the graph of its derivative in 1 IV Clive reasons for your choices A VA A c 111 0 VA d
Calculus
Differentiation
f x In x 2 X
Calculus
Differentiation
The value of sin tan x cot sin sin 1 x COS cot 1 X X is equal to