Calculus Questions

If f x 9x x find f 1 and use it to find an equation of the tangent line to the curve y 9x x at the point 1 8 Need Help Submit Answer 10 0 11 Points Find f a f a f t Read It Watch It DETAILS SCALCET9 2 XP 7 012 MI 8t 60 t 9

d 24 Show that sec x dx sec x tan x

Use the given graph of f to find a number such that if x 1 then If x 11 0 2 8 1 2 1 0 8 f 0 7 1 1 1 X

Use the graph of g to find the value of each expression If an answer does not exist enter DNE y a b d lim g x e X 0 c lim 09 x lim g x x 0 lim g x X 2 lim g x x 2 f lim g x 4 2 2 N 4 9 X

x PERIO 22x 2 f x tan z 3 sin x 3 cos x

10 22 Points Prove the statement using the e definition of a limit x 5 im 3x 4 5 Given 0 we need Select such that if 0 x 51 6 then 4 3x 5 Illustrate with a diagram y 5 6 DETAILS 5 5 6 Thus lim x 5 5 by the definition of a limit 75 5 5 5 5 5 x 5 6 5 Select But 4 x 5 x 1 x 51 x 51 Select v So if we choose 6 5 5 8 5 6 X y 5 6 O 5 5 6 75 5 5 y 5 5 choose Select then 0 x 51 5 5 5 6 5 6 9

Prove the statement using the definition of a limit lim x 0 x 0 Given 0 we need Select such that if 0 Ix 01 Nood Hola Select then Ix 01 Select We have x x Ixl Choose e Then 0 x 01 8 x 01 Select By the definition of a limit I

Let f x 1 x x a Estimate the value of the limit lim 1 x x to five decimal places Does this number look familiar x10 b Illustrate part a by graphing the function y 1 x x 15 10 5 Need Help 10 5 y 10 15 5 10 15 no 15 10 5 15 10 5 5 10 15 5 10 15 X O 15 10 5 16 10 5 10 15 5 10 15 X 15 10 5 y 15 10 5 5 10 15 MY 5

Given that lim 4x 2 6 illustrate this definition by finding the largest values of that correspond to 0 5 0 1 and X 2 0 5 0 1 E 0 05 Need Help Submit Answer 8 S 8 S 8 Read It 1 DETAILS Master It 10 0 23 Points How close to 2 do we have to take x so that the following is satisfied Give the largest possible value 10 000 SCALCET9 2 4 041 MI

Values of f x y are shown in the table below 313 3 3 6 3 4 8 10 13 y 4 5 10 12 16 y 5 12 14 7 Let R be the rectangle 3 3 6 4 y 5 Find the values of Riemann sums which are reasonable over and under estimates for f f x y dA with Az 0 3 and Ay 0 5 over estimate 1 under estimate

For the function A whose graph is shown state the following If the limit is infinite enter oo or co as appropriate If the limit does not otherwise exist enter DNE a lim A x b lim A x X 27 lim A X X 2 d lim w 0 e The equations of the vertical asymptotes Enter your answers X

lim f x 6 lim f x 4 lim x 4 5 5 1 X 5 57 3 Need Help Submit Answer y lim 6 2 Read it Watch It DETAILS 5 X O 5 y 6 4 2 5 X 5 y 6 4 4 0 1 Points SCALCET9 2 2 025 Use a table of values to estimate the value of the limit correct to three decimal places If you have a graphing device use it to confirm your result graphically sin 58 2 5

Evaluate the function f x at the given numbers Round your answers to six decimal places f 9 5 f 9 1 f 9 05 f 9 01 f 9 005 f 9 001 f 8 9 E f 8 95 f 8 99 f 8 995 f 8 999 f x x 9x x 9 5 9 1 9 05 9 01 9 005 9 001 8 9 8 95 8 99 8 995 8 999 x x 72 Guess the value of the limit of f x as x approaches 9 Round your answer to six decimal places If an answer does not exist enter D x2 9x lim x 9 x x 72

a lim f x x 1 1 0 1 Points For the function f whose graph is given state the value of each quantity if it exists If an answer does not exist enter DNE b c lim f x x 3 lim f x x 3 d lim f x x 3 DETAILS e f 3 ya 4 2 0 SCALCET9 2 2 005 2 4 X

For the function of whose graph is shown state the following If the limit is infinite enter co or co as appropriate If the limit does not otherwise exist enter DNE a lim f x 7 b lim f x c lim 144 2 4 6 2 x 0 f x Figure d lim f x x 6 Description x

7 0 1 Points A graphing calculator is recommended a Find the vertical asymptotes of the function y xx DETAILS SCALCET9 2 2 042 10 x 2 7x 4x b Confirm your answer to part a by graphing the function A graphing calculator is recommended y 10 1 10 10 Enter your answers as a comma separated list If an answer does not exist enter DNE 10 y 10 5 5 10 5 X 10 10 5 y 10 5 10 5 10 X 10 5 y 10 10

1 0 1 Points Use a table of values to estimate the value of the limit correct to three decimal places If you have a graphing device use it to confirm your result graphically 1 p lim p 1 1 p Need Help DETAILS SCALCET9 2 2 024 Submit Answer 2 0 1 Points 3 Read It DETAILS Determine the infinite limit Watch It SCALCET9 2 XP 2 015

6 0 1 Points DETAILS A patient receives a 150 mg injection of a drug every 4 hours The graph shows the amount f t of the drug in the bloodstream after t hours f t 300 150 Find lim f t and t 12 t 12 lim f t lim f t 6 12 4 8 lim f t t 127 mg 19 2 2 0 10 MI mg 12 16

Note You can click on the graph to obtain a larger version in a new browser window The graphs of the function f given in blue and g given in red are plotted above Suppose that u x f x g x and v x f x g x Find each of the following u 1

Find an equation for the line tangent to the graph of at the point 1 f 1 f x 4x 5

Let f x 3x sin x cos x Find the following 1 f x 2 f

0 13 Points DETAILS Find the limit If the limit is infinite enter co or co as appropriate If the limit does not otherwise exist enter DNE lim 25x x x 5

Find the equation of the line that is tangent to the curve at the point T 6T y 6x cos x The equation of this tangent line can be written in the form y mx b where and b

Find the limit Let q and r represent arbitrary real numbers If the limit is infinite enter co or co as appropriate If the limit does not otherwise exist enter DNE lim x x qx

Let f x x mmmmm 5x5 2x 4x Then

Use a table of values to estimate the value of the limit If you have a graphing device use it to confirm your result graphically tan 9x lim x 0 tan 4x Need Help Submit Answer 20 0 1 Points Read It lim DETAILS Determine the infinite limit x 6 Master It SCALCET9 2 XP 2 012

Let f x 2T 12T sin T cos z Evaluate f z at z 2

Find the derivative of f x cos I f x

If f t t 6t 6 4t 2 4t 3 find f t

erentiate F f x tan z 3 sin x 3 cos x

Find the derivative of f x x 5 6x 2

a Estimate the value of lim x x 5 x by graphing the function f x x x 5 x Round your answer to one decimal place b Use a table of values of f x to guess the value of the limit Round your answer to one decimal place c Prove that your guess is correct by evaluating lim f x X 00

6 3 4x 3 lim x 00 3x x 3 Need Help Submit Answer 4 0 0 13 Points X Read It DETAILS Watch It Master It PREVIOUS ANSWERS SCALCET9 2 XP 6 007 Find the limit If the limit is infinite enter co or co as appropriate If the limit does not otherwise exist enter DNE t t lim t 6t t

Find the limit If the limit is infinite enter co or co as appropriate If the limit does not otherwise exist enter DNE lim x x X110

0 13 Points A tank contains 9 000 L of pure water Brine that contains 10 g of salt per liter of water is pumped into the tank at a rate of 25 L min a Find the concentration of salt after t minutes in g L c t DETAILS g L b What happens to the concentration in g L as t o g L

X y y x x4 and estimating the asymptotes Enter your answers as comma separated lists If an answer does not exis

lim arctan e 818 Need Help Submit Answer Read It 8 0 13 Points Find the limit If the limit is infinite enter co or co as appropriate If the limit does not otherwise exist enter DNE DETAILS SCALCET9 2 6 037 5 ex lim X 5 6e

2 0 13 Points Sketch a graph of a function f that satisfies all of the given conditions lim f x 00 lim f x 0 lim f x 0 O DETAILS SCALCET9 2 XP 6 002 Need Help y Read It x 1 Watch II lim f x 00 lim f x 00 x ot X 0 y X 1 X 1 X 1

Find the horizontal and vertical asymptotes of the curve You may want to use a graphing calculator or computer to check your work by graphing the curve and estimating the asymptotes Enter your answers as comma separated lists If an answer does not exist enter DNE y 4x x 2 x x 6

Let P and Q be polynomials with positive leading coefficients Consider the limit below If the limit is infinite enter o or o as appropriate If the limit does not otherwise exist enter DNE P x lim a Find the limit if the degree of P is less than the degree of Q b Find the limit if the degree of P is greater than the degree of Q

2 0 N 2 4 1 1 N 2 6 10 3 162 8 a Identify the parent function of the graph depicted in the image 10 bowered by desmos

Find the limit If an answer does not exist enter DNE x 9x 20 X 4 lim X 4

Consider the following functions and calculate their limits as x a numerically by computing the values of the functions near x a Make sure you compute the limit from both sides of a and include a table showing your calculations f x cos x 1 5 f x x a 0 X 2 x x 21 tan 2x 4 x x 2 a 2 Let f x 5 and estimate f 2 numerically by using definition of derivative for small enough values of h Consider the function f x e 2x In x 5 and find the equation for the line tangent to f x at x 0 Plot both f x and the tangent line in one graph to show that your answer is correct Position of an object as a function of time is given by x 3t 2t 5t 1 cos 2t Find velocity and acceleration of this object Plot position velocity and acceleration of this

The figure shows the power consumption in San Francisco for a day in September P is measured in megawatts t is measured in hours starting at midnight P MW 4 800 600 400 200 0 3 6 9 12 15 18 21 a What was the power consumption at 6 00 A M MW What was the power consumption at 6 00 P M MW b When was the power consumption the lowest t h c When was the power consumption the highest t h t h d Find the net change in the power consumption from 9 00 A M to 7 00 P M MW

6 0 167 3 0 333 A 2 0 2 2 1 4 1 Identify the parent function of the graph depicted in the image L 1 6 powered by desmos

The graph of a function h is given range 6 5 4 3 2 1 1 2 3 X 654 IT 321 a Find h 2 h 0 h 2 and h 3 h 2 h 0 h 2 h 3 4 e Find the net chan 45 5 1 2 b Find the domain and range of h Enter your answers using interval nota domain 3 d Find the values of x for which h x 4 O 2 4 and 3 O 2 4 O 3 2 4 O 3 2 and 4 O 3 2 4 5 6 c Find the values of x for which h x 4 Enter your answers as a comma X

The graphs of fand g are given Use them to evaluate each limit if it exists If an answer does not exist enter DNE y y a lim f x g x b lim f x g x c lim f x g x f x x 3 g x d lim e lim lim x x x 2 Fl 13 y f x X y g x e

A graphing calculator is recommended Use the Squeeze Theorem to show that lim x2 cos 22 x 0 x 0 Illustrate by graphing the functions f x x g x x cos 22mx and h x x on the same screen Let f x x 2 g x x cos 22mx and h x x2 Then 2 cos 22x s x cos 22mx Since lim f x lim h x by the Squeeze Theorem we have lim X

Let x 1 f x x a Find the following limits If an answer does not exist enter DNE lim f x lim f x b Does lim f x exist O Yes No c Sketch the graph of f y 1 0 x 1 if x 1 x 2 if x 2 1 0 5 www 2 2 0 5 1 0 1 5 2 0 2 3 4 O y 1 3 1 3 y 4 1 0 0 5 0 5 1 0 1 5 2 0 2 3 4 1 DO y 2 1 2

11 0 11 Points The signum or sign function denoted by sgn is defined by 1 if x 0 if x 0 if x 0 a Sketch the graph of this function y O sgn x DETAILS SCALCET9 2 3 049 iii lim sgn x 0 1 2 11 lim sgn x 310 4 2 2 4 2 b Find each of the following limits If an answer does not exist enter DNE 1 lim sgn x x 0 4 y 4 2 4 2 y 4 2 2 2 4 X 4 2 y 4 2