Differentiation Questions and Answers

24 g v 2 cos v 3 1

f t 2t 4 3 t F

Find the derivative of the function f t 4 tant f t

Find dy dx by implicit differentiation 2x 5xy y 5 y

Differentiate f v f v V v d

Suppose that f 5 1 f 5 4 g 5 6 and g 5 9 Find the following values a fg 5 b f g 5 c g f 5

Find the derivative of f x 1 9x x x in two ways Use the Product Rule f x Perform the multiplication first f x Do your answers agree Yes O No

If H x x cos x find H x and H x H x H x

Differentiate y y t t 1

Differentiate f x f x sec 2 8 secx

Differentiate y y 26 3 25

Differentiate y y x 6 x x 2

Differentiate f t f t 4t 4 t

3 Inverse Values 3 pages max 10 pts Find all possible complex values associated with w 2 sin z at z 3 using the quadratic solve approach Identify which of these values corresponds to the value obtained from wolfram a Evaluate the derivative w 3 Do the same for the value of tanh 3i

H t 10 9 8 6 5 4 if 1 0 if t 0 dom rang

EXAMPLE 2 Velocity and acceleration Suppose the position in feet of an object moving horizontally along a line at time t in seconds is s 51 for 0 1 5 S Figure 3 42 2 2 1 2 3 4 5 Position s 2 5t

3 Use a graphing utility to approximate the real solutions Round to two decimal places Son f x x 4x 2

5 So far we have been using an artificial example because we could calculate the actual value of 5x 3x 1 dx just by using the Fundamental theorem of calculus and finding an antiderivative Now we use a more motivated example 2 a Now consider 1 9x dx Does this function have an 0 elementary antiderivative i Answer ii Note this integral arises when trying to find the length of 3 the curve y x as x goes from 0 to 2 b Now try to estimate the integral of that function from a 0 to b 2 using Riemann sums then record how many subintervals you needed to convince you that your error is less than 001 In parts b c d you will have to experiment with values of n until you are convinced your answer is within 0 001 of the actual value i Answers c Repeat this process but using the trapezoidal rule now How many subintervals were necessary to reach this level of accuracy i Answer d Finally do this again with Simpson s rule but remember to only use even values of n for Simpson s rule i Answer e Of these methods which one converged the fastest Did this result surprise you Share your thoughts i Answer f In each of these cases you also did not have a means of knowing the exact value of the integral so how did you determine when your error was less than 001 Answer

For a function f x we know that f 19 65 and f 19 2 Estimate f 20 f 18 and f 25 f 20 f 18 25 IM Mi

af f x f x dx for the function f x graphed below Hint No calculus is necessary just as we used integrals to find areas can use areas to find integrals The curve shown is a quarter circle Find 5 285 y 2 1 1 2 x square units

1 In a right triangle tan is equal to 1 sine L 2 In a right triangle the relaion hypotenuse opposite adjacer

Evaluate if possible the six trigonometric functions of the real number t If not possible enter IMPOSSIBLE 7 6 sin t cos t tan t t m csc t sec t cot t

Each time your heart beats your blood pressure increases then decreases as the heart rests between beats A certain person s blood pressure is modeled by the function p t 115 25 sin 160nt where p t is the pressure in mmHg millimeters of mercury at time t measured in p t minutes Find the amplitude and period of O O amplitude 25 period O n amplitude 115 period 80 amplitude 115 period amplitude 25 period 80 10 80 H 80

Listen State the amplitude A and period T of the trigonometric function f x 3 sin x OA 3 T T OA T 3 OA 3 T 2 A 3 T 4T 1

Find the value of the expression sin 0 4 02 12 3 1 2 2

Find angle LA of a right triangle if c 25 and b 14 061 034 29 56

Find the radian measure that corresponds to the given degree measure 315 O a 56700 O b 7 n 4 c 7 T

Find the exact value of Cos 0 given that sin and tan 0 0 a 2 14 2 14 5 O b 2 14 2 14 9 c d 4 2 14 9

18 A box with a square base and open top must have a volume of 32 000 cm Find the dimensions of the box that minimize the amount of material used

8 1 2 3 4 5 6 7 8 9 3 f x 4 2 247622 g x 3 92 9 5 3 27 6 Use the table to evaluate the expression fog 1 help numbers

Matching functions with derivatives Match graphs a d of functions with graphs A C of their derivatives YA Functions 2 Derivatives 2 1 2 2 YA b C B A a d 2 X 2 x

Find a formula for the nth term of the sequence in terms of n 1 2 22 23 24 10 15 20 25 30 an for n 1

2 3 2 Sin 5 7 Fo 3

n 0 To cos 0 1

A cost function is given by C x 5 2x 100 6 What is the correct form for the marginal cost Hint You may want to make use of the fact that d ax b an ax b 1 dx Select the correct answer below O MC 60 2x 100 5 O MC 30 2x 100 5 O MC 30 2x 100 5 OMC 60 2x 100 5

Graph several level curves of the following function y Draw the level curves for z 1 z 2 z 3 and z 4 Choose the correct graph below OA OB OC ROO Q 5 Q OOS 27 O OU OD

3 Given an equation coty 4x 0 5y x use implicit differentiation to find dy a

Find the derivative of w x 1 x 6x 4x x 1

39 Given y f x x In x find dy dx

tan x cotr 1 6 Given y f x find dy dx

3 Cubmit Answer 0 se domain of the function is all the x aswers using interval notation Need Help Read It values of the points on the graph and the range is all the corresponding y values From the graph off we see that the domain of f is the interval 00 00 and the range of f is the interval 00 00 x Enter your

all 15 0 0 P bo Free Response Questions 7 Line is tangent to the graph of y x 7 the figure above b Write an equation for line a Find the x coordinates of point P 2018 R 9 To l x y x fe 120 K atea or at 3650 120 60 30 mih alf DIN of 10orpor am bab in2 ung ad to col foldi mo s die win at the point P and intersects x axis at 15 0 as shown in 120 If the line of symmetry for the curve v x x 60 37 120 0 line of symmetry olid goal solomatin y kx b toshod s and age aft et tu tolnimi neb siit zd foblev al snil basyas intersects line at point R what is the length of Ona

Differentiate the function f x 2 0 f x Enter an integer or decimal number more

Show that the curve y 7x 7x 2 has no tangent line with slope 1 y 7x 7x 2 m y for all x Question Help Message instructor but Select an answer 0 for all so m

Find d106 da 106 sin x

Ques Find an equation of the tangent line to the curve at the given point y 3x x 1 2 y Illustrate by graphing the curve and the tangent line on the same screen O 5 13 2 1 8 6 4 2

C Differentiate the function R a 5a 1 R a

Differentiate the function 3 F x F x 5 I 10

Differentiate the function S R 8TR S R

Differentiate the function h x x 2 3x 4 h x