Differentiation Questions and Answers

A rechargeable battery is plugged into a charger The graph shows C t the percentage of full capacity that the battery reaches as a function of time t elapsed in hours C percentage of full charge 100 80 60 40 20 2 4 6 b Sketch the graph of C t 8 10 12 t hours a What is the meaning of the derivative C t O C t is the total percentage of full capacity with respect to elapsed time in hours O C t is the average rate of change of percentage of full capacity with respect to elapsed time in hours O C t is the instantaneous rate of change of percentage of full capacity with respect to elapsed time in hours O C t is elapsed time in hours required for a given percentage of full capacity to be reached C

a Find the slope of the tangent line to the parabola y x 7x at the point 3 12 by using the following parameters i The tangent line to the curve y f x at the point P a f a is the line through P with slope m lim x a m ii The expression of slope is m lim h 0 y m b Find an equation of the tangent line in part a 10 5 20 c Graph the parabola and the tangent line As a check on your work zoom in toward the point 3 12 until the parabola and the tangent line are indistinguishable y y y 5 5 5 y 10 15 10 10 20 f a h f a h V 5 10 X f x f a x a 10 provided that this limit exists X 1 10 5 5 SE 10 15 5 X

Trace or copy the graph of the given function f y V Sketch the graph of f on the same coordinate axes y

A curve has equation y f x a Write an expression for the slope of the secant line through the points P 2 f 2 and Q x f x O 2 2 f x X f x f 2 X 2 X 2 f x f 2 f x X f 2 2 b Write an expression for the slope of the tangent line at P X 2 X 2 f x f 2 Olim lim X 2 f x f 2 X 2 O lim f x f 2 lim X40 X 2 x 0 2 2

For the function f x y z 3x 2y 2 find f fy and f e 4 Type an exact answer using radicals as needed

2 f x y z 3x 3y z x Type an exact answer using radicals as needed

dy Find by implicit differentiation dx 19xe 20 dy

for the equation xy 5 using implicit differentiation dx Find

The curves r t 4t t t and r t sin t sin 2t 5t intersect at the origin Find their angle of intersection 8 correct to the nearest degree 8 75 x

Verify that wxy Wyx for w In 5x 8y

of af For the function f x y sin 8x 4y find and ax dy of

Find by implicit differentiation dy dx x5 y5 27 dy 11

14 Find the length and width of a rectangle that has perimeter 48 meters and a maximum area

2 A conical tank with vertex down is 18 feet across the top and 22 feet deep If water is flowing into the tank at a rate of 18 cubic feet per minute find the rate of change of the depth of the water when the water is 6 feet deep V r h 3 see text section 2 6

Find by implicit differentiation dy dx 10xey 7 dy dx

dy Find by implicit differentiation dx 7 x 8 y 21 dy

dy Find at the given point dx 5x y 8 x 8y at 0 1 dy The value of at 0 1 is dx

Find dy dx X at the given point 3 5 at 1 1 The value of dy dx at 1 1 is

Find by implicit differentiation dx 11x 13 ey Hy

Find the limit Let g and h represent arbitrary real numbers If the limit is infinite enter or lim x gx x hx

dy Find for the equation 7x 4y 1 using implicit differentiation dx dy dx

Find dy dx by differentiating implicitly When applicable express the result in terms of x and y 6x 5y 7 dy dx II

farmer has 1800 m of fencing and wants to enclose a rectangular field bordering a river with no fencing needed along the river Let x represent the width of the field Complete parts a throu below a Write an expression for the length of the field Let the length of the field be the distance along the river x 0

The graph of g is given below Use the graph of g to determine the derivative of g at x 10 9 8 7 6 5 4 3 2 11 12 2 10 9 8 7 6 43254 21 S 2 234 4 567 8 9 10 1 2 3 4 5 6 7 8 9 10 11 2

If 0 is in the third quadrant and sin 8 5 what is cos 0 3 Determine your answer without using a calculator Rationalize the denominator if necessary cos 1 0 Enter a or sign Enter

For the following exercises find in polar form 29 21 21 cis 135 3 cis 65 30 2 cis 90 2 cis 60 15 cis 120 3 cis 40 6 cis 2 2 cis 7 33 2 5 2 cis 2 2 cis 2 cis 5 2 3 cis 4 22 31 21 32 21 34 21

7X3 Step 1 dy 9 21x 0 7X ay ax 7x32 Step 2 0 189 X 7x332 Step 3 27 7x4 189x 7832

Test if the following series is conditionally convergent absolutely convergent or divergent 6 Test if the following series is conditionally convergent absolutely convergent or divergent o a n 1 o 1 n Inn b tan 1 n n 1 c 1 n 1 1 1 2 n

Use implicit differentiation to find dy dx www y cos xy 2y x cos xy Question Help T dy dx Massage instructor given the equation sin xy y X

The graphs of f solid and g dashed are given below Let h x f g x and k x g f x 6 5 43 2 1 16 4 3 2 1 1 3 4 5 5 6 q Determine h 1 Enter DNE if the derivative fails to exist Determine k 1 Enter DNE if the derivative fails to exist

Draw the following graph on the interval 345 x 0 y cos x 170 200 100

Find the value of the derivative if it exists at the indicated extremum If an answer does not exist enter DNE 2 10 3 9 f x 5x x 1 2 1 1

5 We define the Laplace Transform of a function f t as follows f t est f t Use the above definition to prove that at f t dt c t 1 2 Hint Use integration by parts Set u t and du etdt In your computations treat s a constant

0 89 Points DETAILS Consider the following graph Domain 6 Range 4 2 y 6 4 2 41 2 4 State the domain and range of the graph in interval notation Simplify your answers 6 Determine whether the relation represents a function Yes it is a function No it is not a function X

Question 6 If f u f u 4 u Cos v v2 dv then

Sketch the graph of a function f that is continuous on 1 8 and has the given properties Absolute maximum at 3 absolute minimum at 8 7 is a critical number but there is no local maximum or minimum there y 10 8 nn 6 y 12 10 8 12 10 8 6 N 00 y 12 10 8 6 N 3 10 8 67 8

2 Is the piecewise function CONTINUOUS Is the function DIFFERENTIABLE everywhere over its domain Why or why not Explain in words and include algebraic justification Hint Consider all areas in domain of interesting behaviour of your piecewise function This explanation must also be included in the video recording to be submitted with this part of the project These are the 2 functions in my piecewise function f x 5 x 7 2x 2 2 g x sin3 7 21022

Use separation of variables with 16 to find a product solution to the following partial differential equation du 0 that also satisfies the conditions 0 0 9 and 0 0 8

5 Using the 2nd Derivative Test given below find the local extremas or saddle point of f x y x 4xy y THEOREM 11 Second Derivative Test for Local Extreme Values Suppose that f x y and its first and second partial derivatives are continuous throughout a disk centered at a b and that f a b f a b 0 Then a b C d e i f has a local maximum at a b if fx 0 and ffy f 0 at a b ii f has a local minimum at a b if f 0 and ffy f 0 at a b iii iv f has a saddle point at a b if ff fy 0 at a b The test is inconclusive at a b if fm 0 at a b In this case we must find some other way to determine the behavior of fat a b P 0 0 is local minimum point P 1 1 and P 1 1 are saddle points P 0 0 is local maximum P 1 1 is local minimum and P 1 1 is saddle point P 0 0 is saddle point P 1 1 and P 1 1 are local minimum points P 0 0 and P 1 1 are local minimum points P 1 1 is saddle point P 0 0 is saddle point P 1 1 is local minimum and P 1 1 is local maximum point Bo b rak

1 Let F t cost tsint i a Use the formula K sint t co 1 dr dt to find the curvature dll 15p ADEL

3 Let f x y xy 1 xy a Find linearization L x y zo f P x x f P y y at the point P 1 2 2 3 M b Find an upper bound of the error E x 11 y 2 in f x y L x y approximation over the rectangle R

2 a Let f x y arctan x y and Duf P Vf P Find the direction cos 0 7 sine for which Duf 0 1 2 15p

Cost revenue and profit rates Suppose that for a com pany manufacturing calculators the cost revenue and profit equations are given by C 90 000 30x R 300x 30 P R C where the production output in 1 week is x calculators If production is increasing at a rate of 500 calculators per week when production output is 6 000 calculators find the rate of increase decrease in C 72 000 60x R 200x A Cost B Revenue C Profit Cost revenue and profit rates Repeat Problem 33 for x 30 P R C where production is increasing at a rate of 500 calculators per

Problem 48 Use the price demand equation to determine whether demand is elastic inelastic or has unit elasticity at the indicated values of p x f p 1 875 p

The radius of a spherical balloon is increasing at the rate of 3 centimeters per minute How fast is the volume changing when the radius is 10 centimeters Refer to Problem 21 How fast is the surface area of the sphere increasing when the radius is 10 centimeters

Find the derivative r t of the vector function r t tan 8t sec 7t tan 8t 1 2 r t

Match the equation with its graph x 6z 1 x ix y

Find the marginal revenue function R x 2x 0 06x

For the following quadratic complete the square to turn it into its vertex form f x 2x 12x 6

Consider the equation f x x 4 7 Identify the parent function and describe ALL the transformations shifts stretches etc 3