Differentiation Questions and Answers

Let F x 1 lim F 2 lim F 2 and I 1 I 1 both 2 Find the following limits O both 1 O2 and 1 O2 and 2

Find f x and f x of the function f x 5 cos x sin x Select 2 correct answer s Of x 5 sin x sin 2x Of x 5 cos 2x 2 cos x Of x 5 cos x 2 cos 2x Of x 5 sin 2x sin x Of x 2 cos 2x 5 cos x

Solve the triangle m 35 a 125 C

Given vectors a 5 3 1 and b 1 3 2 calculate the following 1 3a b II a b III a x b IV proja b

Solve the equation for x det 1 2x 4 0 9 4 5 7 x X 61

Using the vectors shown in the figure write each sum or difference as a single vector A a AB BC b CD DB c DB AB D d DC CA AB C B i

The graph off is shown below For which values of x is f x zero 6 F 15 10 5 5 2 4 6 X

I want to build two gardens within a rectangular field as shown below I have 800 feet of material with which to build all of the fencing for the gardens What dimensions should I use to make sure my gardens are as large as possible

0 0 1 0 1 0 1 0 1 0 State what the shaded area represents A Distance traveled in miles B Acceleration in miles per hour per hour C Position in miles from starting point D Final velocity in miles per hour 4 DE Next question Int s possible Velocity miles per hour Time hours

Determine whether the Mean Value Theorem can be applied to the function f x x on the closed interval 0 2 If the Mean Value Theorem f 2 f 0 be applied find all numbers c in the open interval 0 2 such that f c 2 0 MVT applies 2 3 3 O MVT applies 0 5 MVT applies 2 3 O MVT applies 1 O MVT does not apply

1 point a Find an angle between 0 0 and 0 360 that has the same sine as sin 152 but is not 152 0 has the same sin as sin 152 b Find an angle between 0 0 and 0 360 that has the same cosine as cos 231 but is not 231 has the same cos as cos 231 Note Do not include units in your answers Just give the numerical values 0

Tutorial Exercise Use the properties of summation and the Summation Formulas Theorem to evaluate the sum Use the summation capabilities of a graphing utility to verify your result Step 1 24 9i i 1 To evaluate the sum Step 2 1 9i the following property of summation can be used to find the sum n n ka k a i 1 i 1 Using the above property 9 n 24 1 24 Use the Summation Formulas Theorem to find 9 Recall that Theorem 5 2 states the following i 1 24 9 i i 1 n n 1

The voltage V in a simple electrical circuit is slowly decreasing as the battery wears out The resistance R is slowly increasing as the resistor heats up Use Ohm s law V IR to find how the current I dv dR is changing in A s at the moment when R 345 2 I 0 03 A 0 02 V s and 0 08 2 s Round your answer to six decimal places dt dt A s

Use sigma notation to write the sum 2 j 1 Need Help Read It 6

z Use the chain rule to find and s z s z at z tan az at u 4s 7t v 7s 4t

2 5 h t NA 0 5 5 2m Using the image above find a sine or cosine equation for the function h t

Q14 Determine f 4 from the following table X f x 1 0 2 1 4 45 8 21 10 27

Find the differential of the function dz z e 7x cos 2 t dx C 1 dt

Use differentials to estimate the amount of tin in cm in a closed tin can with diameter 10 cm and height 18 cm if the tin is 0 04 cm thick Round your answer to two decimal place cm 3

Use the equation to find dy dx dy dx dy dx F x F y F 3y cos x x y

Show that the function f x y xy 8y2 is differentiable by finding values of and 2 that satisfy the following definition Definition If z f x y then f is differentiable at a b if Az can be expressed in the form Az f a b Ax fy a b Ay E AX E2Ay where and 2 0 as Ax Ay 0 0 To find an expression for Az we choose an arbitrary point a b another point arbitrarily close to the first a Ax b Ay and compute the difference between their respective f values Az f a Ax b Ay f a b a Ax b Ay 8 b Ay ab aAy b x AxAy 8b2 16bAy 8 Ay 2 ab Ay bAx AxAy but fx a b and f a b Az f a b Ax f a b Ay AxAy 8AyAy Let Ay and 2 8Ay therefore f is differentiable by the given definition AyAy SO

Consider the following 1 y 6 5 4 3 2 1 0 5 1 2 1 1 1 2 3 4 a Find the least squares regression line y 5 0 5 6 X b Calculate S the sum of the squared errors Use the regression capabilities of a graphing utility to verify

a Approximate the area under graph a of f x 2 x over the interval 1 9 by computing the area of each rectangle to four decimal places and then adding b Approximate the area under graph b of f x 2 x over the interval 1 9 by computing the area of each rectangle to four decimal places and then adding Ay 2 1 0 Ay 2 1 1 0 1 2 3 f x 21x 5 x 27x1 3 4 5 7 6 7 80 a 9 b 9 X X Q Q C The area under graph a is approximately Round to four decimal places as needed

ASK YOUR TEACHER PRACTICE ANOTHE The wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed Values of the function h f v t are recorded in feet i the table below Wave height as a function of wind speed and duration Duration hours V 10 15 20 Wind speed knots 30 40 50 60 t 5 2 4 10 15 2 4 5 7 2 20 30 8 2 2 55 8 40 54 5 What is the value of f 30 20 What is its meaning According to the table f 30 20 9 40 50 45 2 2 9 13 16 17 14 21 25 28 31 33 33 19 29 36 24 37 62 5 9 5 9 18 19 19 48 50 67 69 which means that if a 30 knot wind has been blowing in the open sea for 20 hours it will create waves with estimated heights of

f x x 7 Plot four points on the graph of the function the leftmost point and three additional points Then click on the graph a function butto

Find the first partial derivatives of the function aw x w y w W In x 4y 6z

1 For this problem consider the function f x y 3y x 18yx x a Find the critical points of f

Find the first partial derivatives of the function w du W ev u v4

Find the first partial derivatives of the function f x y x7y fx x y f x y

Find the indicated partial derivative 8 f x y y sin xy 4 4 4

The figure shows the graph of a function At the given value of x does the function appear to be differentiable continuous but not differentiable or neither continuous nor differentiable 18 x 0 4 4 2 4

A company s total cost in millions of dollars is given by C t 80 70e graph of C t is shown to the right Find each of the following a The marginal cost C t b C 0 c C 4 d Find lim C t and lim C t t a C t Do not include the symbol in your answer t where t is the time in years since the start up date The Ay 150 75 To 15 30

a U 3 d w 2 dz w 5z ez Find for the given function

Solve the right triangle shown in the figure Ro B A C b a DO B 33 b 50 3 A 57 a 77 5 c 92 4 A 33 a 32 7 c 92 4 A 33 a 32 7 c 42 2 A 57 a 77 5 c 60

Use the given triangles to evaluate the expression Rationalize all 2 45 1 45 1 60 2 1 30 3

Differentiate f w f w SCALCET9 3 3 017 1 sec w 1 sec w 2 sec w tan w 1 sec w 12

Let f x x 1 Click on the graph where the corresponding points are located for the X values of 4 3 2 1 0 1 and 2

3 4 018 MI Find the derivative of the function f t 7t sin t f t 7ut cos nt 7 sin

Differentiate f t f t 333 Vt t 3 t 3 t t 3 2

Differentiate the function 5 x 4x 2 3x y 1 y 4 3x 5 2x 5 2

Differentiate the function g x g x 1 x x x

values of c in the open interval 0 8 such that f c 0 3 Let f x sin Find all values of c in the interval 1 0 such that f c 2 0 Let f x cos 4 Find 11 T 3TT

Let f u u and g x u 4 12x Find f o g 3 fog 3 Type an exact answer

Sketch a graph of f x 3 2 1 5 4 3 2 1 1 2 3 4 5 1 if x 1 2x 1 if 1 x 1 if x 1 2 2 3 4

Express the number 3 3 6i in the form R cos 0 i sin 0 Rete where R 0 and 0 T NOTE Round the value of 0 to exactly two decimal places 3 3 6i

Use the given graph to determine the open intervals on which the function is either increasing decreasing or constant Enter solutions using interval notation If no interval exists where the function is either increasing decreasing or constant enter DNE 10 9 8 7 6 10 9 8 7 6 5 4 3 2 1 54 3 2 2 3 4 5 597 6 7 8 9 10 3 4 5 6 7 8 9 10 The function is increasing on the interval s The function is decreasing on the interval s The function is constant on the interval s

The graph of f is given below Identify the intervals on which f is increasing or decreasing Identify the critical values and the location z values of any relative extrema of f Use interval notation to report interval solutions If multiple relative maximums or relative minimums exist enter those solutions using a comma separated list If a solution does not exist enter DNE 5 75 3 2 4 6 3 45 2 3 1 15 1 15 2 3 3 45 5 75 f is increasing on f is decreasing on The critical values of f are f has a relative maximum at x 2 3 9

In the following triangle the lengths of the sides are NOT drawn to scale and the letter Q is used to represent 8 Q 9 8 For the triangle given above express the length of s using a trigonometric function of the angle 0 Os 9 8 sin 0 Os 9 8 cos 0 Os 9 8 tan 0 Os 9 8 cot 0 Os 9 8 sec 0 8 9 8 csc 0

4 Grain is pouring from a chute at the rate of 10ft min forms a conical pile whose height is always twice its radius How fast is the height of the pile increasing at the instant when the pile is 2ft high

Identify the exponential function whose graph is given below Provide your answer below n 14 21 2 3