Calculus Questions

dr VI dx evit got O A O B 3r e 1 e x 2 e r 2 1 e r 1 2 OC 1 1 2 c O D e3x 3 2 1 e z E 3 1 e a 1 2

The average value of v t et cos t over the domain 1 4 is approximately OA 2 840 105 OB 2 516 x 105 O C 1 149 x x 106 OD 1 688 OE 0 562

If fis continuous for all x which of the following integrals necessarily have the same value cb a f x dxII f x a da III cb 1 f x d ll I 0 ab c 1 f x c dx a c OA I and II only B I and III only OC II and III only D I II and III E No two necessarily have the same value

dlog5 x 1 dx O A OB B O C Hol O D 1 2 x 1 In 5 2 2 1 In 5 x 2 x 1 X D 2 In 5 2 1 OE None of the above

f x ln 3x 36 Then f x 24x O A 3 36 OB O C O D 12x In 3x 36 3 x 36 12x 37 36 2 ln 3x 36 3 36 6x O E E 3 36

y g x on 3 4 is given by the following curve which is composed of two line segments and a semicircle Click here for long description Evaluate f3 g t dt OA 2 3 OB 2 3 O C 3 OD 2T 3 OE 6 W

The function g x 12x sin x is the first derivative of f x If f 0 2 what is the value of f 2 O A 2 5 OB 8T2 2 OC 32 2 OD 30T 2 OE 47 8

Solving the equation 16 x OA x 3 5 OB X 2 23607 OC x 1 or x 2 OD x 2 7182818 OE None of the above you get x

Suppose f x 9 x Then on any interval where the inverse function y f x exists d x dx OA 2x 1 B 9 x 1 2 O c 2 9 x 2 OD 9 x OE 2x

Which of the following ould represent an y f x A b where b 1 I II III 2 40T A I only B II only C III only D I and II only 30 20 10 2 x 2 1 20 1 y f x 40 X 2 2 5 5 1 1 4 y f x 2 2 5 5 20 2 40

For all positive x log log4 a Hint Think change of base O A log4 x log x B log4 2 1 2 O c log4 x O D log24 2 O E log x

Consider the following graph of y 10 8 6 4 O A I only OB II only O C III only 2 D Both I and III 10 5 0 5 Which of the following could be the equation for f x I log ax where a 0 II logs ax5 where a 0 III log ax where a 0 Hint Simplify them algebraically first 10

1 Which of the following will give the smallest value for da for a given number of intervals A A right hand Riemann sum approximation B A trapezoid approximation OC The actual value of the definite integral D A left hand Riemann sum approximation E A midpoint Riemann sum approximation

The region Ris bounded by the x axis x 2 and y x Which of these expressions represents the volume of the solid formed by revolving R about the line x 2 OA ST ydy B 2 x dy c f 2 x dy OD ST 2 y dy OE ST 2 y dy

Suppose f x sin cos x On any interval where the inverse function y f x exists the derivative of f x with respect to x is OA B C D E 1 COS cos y where x and y are related by the equation X sin cos y 1 sin y cos cos y x sin cos y where x and y are related by the equation 1 sin y cos COS y 2 X sin cos y 1 cos TC TT COS X equation x where x and y are related by the equation where x and y are related by the equation satisfy the sin cos y 1 sin x cos cos x X sin cos y where x and y are related by the equation

What is the area bounded by y x and y 3x A 5 OB 8 O C OD 25 OE 11 2 72

If f 3x x 1dx udu then So S a A a 2 b 4 OB a 8 b 64 OC a 12 b 48 OD a 7 b 63 O E None of these

4 If f 3x x 1dx udu then 2 a O O A a 2 b 4 B a 8 b 64 C a 12 b 48 OD a 7 b 63 OE None of these

Consider the function f x 100 of the following are true I lim f x 70 XIX II lim f x 70 118 III lim f x 100 18 IV lim f x 100 118 V lim f x 100 x 0 12000 400 3000a 12000 400 3000a OA I and V only B I and IV only OC II and III only OD III and V only OE I IV and V only where 0 a 1 Which

Suppose y f x 3x 5 Then the inverse function f x 3 A 2 5 3 OB 5 5 O c 5 3 O D 3y 5 E 3x 5 3 5

Which of the following statements is true I 3 3 II 24 48 812 III 3 9 2 33x 4 OA I only OB II only OC Ill only OD II and III only OE None of the statements is true

Suppose that y f x x 4x 4 Then on any interval where the inverse function y f x exists the derivative of y f x with respect to x is Hint x 4x 4 can be factored and rewritten as something squared A Both where x and y satisfy the equation y x 4x 4 and x 1 2 B 1 2y 4 1 2x 4 OC 2 4 where x and y satisfy the equation y x 4x 4 2y OD 2 O Ex 1 2

Suppose f x Find the slope of the tangent line to the graph of y f x at 3 1 You get OA 972 1 O B 24 C 00062 D 1620 E 1 16 4x

Suppose x 4x f Find the slope of the tangent line to the graph of y f x at 3 1 You get OA 972 1 OB 24 C 00062 D 1620 E 16 O E

A mass suspended from a spring is raised a distance of 5 cm above its resting position The mass is released at time t 0 and allowed to oscillate After one third of a second it is observed that the mass returns to its highest position which was 4 5 cm above its resting position What is the rate of change of the position of the mass at t 2 1 seconds Round your answer to three decimal places

An isotope of sodium has a half life of 20 hours Suppose an initial sample of this isotope has mass 10 grams The amount of the isotope in grams remaining after thours is given by NOTE We re assuming exponential decay here Use the exponential decay function OA 10 965936329 B Both 10 1 035264924 and 10e 965936329t C 10e 965936329t D 10 1 035264924 t E Both 10 965936329 and 10e 965936329t

Suppose that y f x 2x x 0 Find a c 0 such that the tangent line to the curve y f x at x chas the same slope as the tangent line to the curve y f x at x c You get O A c B c 1 3 12 3 3 2 3 C c D c 1 E c 1 C 12

Suppose 4 O O O O A In 12 OB 3 O c log4 12 OD log12 2 O E 12 en x

Suppose that f x is an invertible function that is has an inverse function and that the slope of the tangent line to the curve y f x at the point 2 4 is 0 2 Then OA The slope of the tangent line to the curve y f x at the point 2 4 is 5 OB The slope of the tangent line to the curve y f x at the point 2 4 is 5 O C The slope of the tangent line to the curve y f x at the point 4 2 is 5 OD The slope of the tangent line to the curve y f x at the point 4 2 is 5 O E The slope of the tangent line to the curve y f x at the point 4 2 is 0 2

X F where F ri yj and S is the curved portion of the right circular cone given by 2 y 2 0 0 1 Hide Transcribed Text b Calculate iint S boldsymbol nabla times boldsymbol F cdot d boldsymbol S where F xi yj and S is the cunod portion of the right circular cong given by

Solving the equation 33x 235 2 for x you get x O A 17 O B log3 5 O c log3 5 OD 7 O E 35

f x x x 18x 4 Over which of these domains could you define f x Remember that f x must be one to one only one y value for each x value over the domain where f x is defined as a function So in some cases you must restrict the domain of f x so that it s one to one There might be more than one section of domain that s one to one HINT The function f x is one to one wherever it s monotonic monotonic simply means increasing or decreasing over the entire interval so the derivative does not change sign To find these intervals start by taking the first derivative which is f x 3x 3x 18 and setting it equal to zero 1 3 II 3 III 2 00 OA Either I or II B Either I or III C II OD I E III

Let f x x 3 2 The graph of the inverse function y f x has a vertical tangent at HINT Think about the symmetry of the 2 graphs O A 10 1 O B 66 1 O C 1 10 O D 2 3 OE 3 2

Graph the function f x over the given interval Partition the interval into four subintervals of equal length Then add to your sketch the rectangles associated with the Riemann sum f ck Axk given that ck is k 1 the a left endpoint b right endpoint and c midpoint of the kth subinterval Make a separate sketch for each set of rectangles f x sinx 4 b Sketch the rectangles associated with the Riemann sum f ck Axk given that ck is the right endpoint of the kth subinterval Choose the correct graph below k 1 O A O E AL OB O F Ay N IN G O C O G Ay Ay Q Q Q O D OH Ay Q

4 k 1 Graph the function f x over the given interval Partition the interval into four subintervals of equal length Then add to your sketch the rectangles associated with the Riemann sum 1 ck Axx given that c is the a left endpoint b right endpoint and c midpoint of the kth subinterval Make a separate sketch for each set of rectangles f x sinx OA E OE c Sketch the rectangles associated with the Riemann sum f ck Axk given that ck is the midpoint of the kth subinterval Choose the correct graph below B 4 O F k 1 E Ay S O C O G Ay n 14 Ay S O D OH DUN Ay 0004 Q

4 Graph the function f x over the given interval Partition the interval into four subintervals of equal length Then add to your sketch the rectangles associated with the Riemann sum f c Axk given that c is k 1 the a left endpoint b right endpoint and c midpoint of the kth subinterval Make a separate sketch for each set of rectangles f x sinx n a Sketch the rectangles associated with the Riemann sum f ck AxKk given that ck is the left endpoint of the kth subinterval Choose the correct graph below k 1 O A 1 O E Q O B 4 O F Ay Q O C O G AY Ay Q O D OH Ay DON

b g b Use the Substitution Formula ff g x g x dx f u du where g x u to evaluate the following integrals g a a y 1 dy b a 0 S y 1dy 1

Which of the following expresses 1 3 9 27 81 243 729 in sigma notation 7 a 3k 1 k 1 6 b 3k k 0 5 c 3k 1 k 1

Find the volume of the following solids The base of a solid is the region between the curve y 8 sin x and the interval 0 on the x axis The cross sections perpendicular to the x axis are a equilateral triangles with bases running from the x axis to the curve as shown in the figure b squares with bases running from the x axis to the curve y 8 sin x

Find the derivative of the given function r t tan 2 5t 11

Suppose that x and y are related by the equation 4x y 4 and use implicit differentiation to determine dy dx dy

Find the derivative of the given function y 8sin 4x 7 dy dx II

Linearize f x 4 x for a 3 and use it to approximate the value of 1 8 x

Find the derivative of the given function R T7e T dR dT

Find the derivative of the given functio 7x 6 x dy y In

Find the derivative of the given function y 6sin 5x 4 dy dx

Suppose that x and y are related by the equation 4x 3y 3 and use implicit differentiation to determine dx dy dy 11

Differentiate the function dy dy y 1 3x 1 5

Find the derivative of the following function 5x 8 f x f x 2 x 4x 9

8 points EACH part Verify the Divergence Theorem is true for the vector field F x y z xi y j zk and t surface S is the plane 2x 2y z 6 bounded by the coordinate planes in octant 1