Calc Final Semester 2

function f is twice differentiable and f'(x) >0 for the interval 0<x<8. if the graph of f is concave downward for all x and passes through the points (3,10) and (6.22), which other point could also lie on the graph f?

(1,)

∫x(3x)^0.5 dx

(2(3)^0.5)/5 x^5/2 +c

find the point on the graph y=√x between (1,1) and (9,3) at which the tangent to the graph has the same slope as the line through (1, 1) and (9,3)

(4, 2)

if the graph of y=f(x) is defined for all x≥0, contains the point (0,1), has dy/dx = 3 √ xy, and f(x) >0 for all x, then f(x) =

(x^3/2 +1)^2

let f be a differentiable function with f(4) = 6 and f '(4)=3. if g is the function defined by g(x) = x/ f(x) , then g '(4) =

-1/6

if f(x)= (sin)^2 / 1-cosx , then f '(x) =

-sinx

lim sin (5+h) -0.5/h h ->0

-√ 3/2

given that x^2 -2x -1/x-1 for x≠1 f(x) = k for x=1 determine the value of k for which f is continuous for all real x

0

a missile rises vertically from a point on the ground 75,000 feet from a radar station. if the missile is rising at a rate of 16,500 ft/ min at the instant when it is 38,000 ft high, what is the rate of change, in radians per minute, of the missile's angle of elevation from the radar station at this instant?

0.175 rad/min

approximate ∫ from 0 to 1 (sinx)^2dx using the trapezoidal rule with n=4 to three decimal places

0.277

if xy^2 = 20, and x is decreasing at the rate of 3 units per second, the rate at which y is changing when y=2 is nearest to

0.6 units/sec

a point moves on the x-axis in such a way that its velocity at time t>0 is given by v= e^t /t . at what value of t does v attain its minimum?

1

if f(x) = x^3 -x +6 for all real numbers x, and if g is the inverse function of f, then f ' (g(0))g'(0) =

1

let f and g be differentiable functions with the following properties: (i) g(x)>0 for all x (ii) f(0) = 1 if h(x) = f(x)g(x) and h'(x) = f(x)g'(x) , then f(x) =

1

If 0 ≤ x ≤ (π/2) and the area under the curve y=cosx from x=k to x= (π/2) is 0.1, then k =

1.12

If ∫ from 0 to 1000 8^x dx - ∫ from a to 1000 8^x dx = 10.40, then a =

1.5

let f(x) = ∫ cotxdx; 0<x<π . if f(π/6) = 1, then f(1) =

1.521

if the definite integral ∫ from 1 to 3 (x^2 +1) dx is approximated by using the trapezoidal rule with n=4, the error is

1/12

if the line tangent to the graph of the function f at the point (3/-2) has its x-intercept at x=7, then f'(3) =

1/2

∫ dx/(9+x^2) dx =

1/3 tan^-1 (x/3) + c

if the function 3ax^2 +2bx +1 if x≤1 f(x) = ax^4-4bx^2-3x , if x>1 is differentiable for all real values of x, then b=

1/4

on the interval [0,6], function f has an average value of 20, but for the interval [0,10], its average value is only 16. what is the average value of f on the interval [6,10]?

10

the average value of sec^2(x) on the interval [π/6, π/4] is

12-4 √ 3 /π

administrators at Massachusetts General Hospital believe that the hospital's expenditures E(B) = 14000 + (B+1)^2 . On the other hand, the number of beds B is a function of time t, measured in days, and it is estimated that B(t) = 20sin(t/10) +50. At what rate are the expenditures decreasing when t=100?

135 dollars/day

the radius of a sphere is increasing at a rate proportional to its radius. if the radius is 4 initially, and the radius is 10 and after two seconds, what will the radius be after 3 seconds?

15.81

the graph g' , the derivative of function g is shown. if g(4)=10, then g(0) =

17

what is the trapezoidal approximation of 3 ∫ (e^x )dx usingn n=4 subintervals? 0

19.972

the sale of lumbar s (in millions of square feet) for the years 1980 to 1990 is modeled by the function s(t) = 0.46cost(0.45t + 3.15) where t is time in years with t=0 corresponding to the beginning of 1980. determine the year when lumbar sales were increasing at the greatest rate

1983

an function f is defined by f(x)=x^3 -3 √ x for the interval 1≤x≤4. During this interval, at what value of x is the instantaneous rate of change of f equal to its average rate of change for 1≤x≤4?

2.641

the maximum acceleration attained on the interval of 0≤t≤3 by the particle whose velocity is given by v(t)= t^3 - 3t^2 +12t +4 is

21

the volume of a cube is increasing at a rate proportional to its volume at any time t. if the volume is 8ft^3 originally, and 12ft^3 after 5 seconds, what is its volume at t=12 seconds?

21.169

the graph of f over the interval [1,9] is shown in the figure above. using the date in the figure, find a midpoint approximation with 4 equal subdivisions for 9 ∫ f(x)dx 1

24

the flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph shown. of the following, which best approximates the total number of barrels of oil that passed through the pipeline that day?

2400

use differentials to approximate the change in the volume of a sphere when the radius is increased from 10 to 10.2cm

25.133

on the ∫ from -1 to 1 on 4/(1+x^2) dx

2π

in the earth's atmosphere the speed of sound is a function of the altitude. the figure above consisting of 3 line segments, shows the speed of sound, s(a), in m/sec as a function of altitude, a, in meters. The graph is not drawn to scale. What is the average speed of sound in m/sec on the interval [0,3200]

304.4

the graph of f is shown in the figure above. if 3 ∫ f(x)dx =2.3 1 and F'(x)=f(x), then F(3)-F(0) =

4.3

an approximation for 2 ∫ e^(sin(1.5x-))dx -1 using right hand riemann sums with three equal subdivisions is nearest to

4.5

the parabola x=5y=y^2 and the line x+2y=10 intersect at the points (0,5) and (6,2). The area of the region bounded by the parable and the line is

4.5

the rate of change, R, of the temperature in a room during a 5-hour interval 0≤t≤5 is given by R(t_ = 4cos(3/t+1). during the period the temperature is rising, the number of degrees it increases is given by the definite integral

5 ∫ R9T0 dt 0.91

the side of a square is increasing at a constant rate of 0.4 cm/sec. in terms of the perimeter, P, what is the rate of change of the area of the square in cm^2/sec?

6.4P

a company manufactures x calculators weekly that can be sold for 75 - 0.01x dollars each. The cost of manufacturing x, calculators is given by 1850 + 28x - x^2 +0.001x^3 . The number of calculators the company should manufacture weekly in order to maximize its weekly profit is

683

the graph of function f consists of two semicircles, as shown above. find ∫ from 0 to 12, (f(x) - 2) dx

6π - 24

if the line y=4x+3 is tangent to the curve at y=x^2 + c , then c is

7

an object moves along the x-axis so that at any time t its acceleration is given by a(t) - arctan (2+t) . if the velocity of the object is t at time t=1, then its velocity at time t=3 is

7.642

as the limit approaches 0, (tan(2x))^3 / x^3

8

find k so that f(x) = (x^2 - 16)/(x-4), if x≠0 is continuous for all x and f(x) = k, if x=4

8

The volume generated by revolving around the y-axis the region enclosed by the graphs y=9=x^2 and y=9-3x, for o≤x≤2, is

8 π

the equation of y=x-3sin(π/4)(x-1) has a fundamental period of

8π

if three subdivisions of [0.3] are used, what is the trapezoidal approximation of ∫ 0 to 3 (x^2 -6x +9)dx

9.5

if the base b of a triangle is increasing at a rate of 3 inches per minute while its heigh h is decreasing at a rate pf 3 inches per minute, which of the following must be true about the area A of the triangle?

A is decreasing only when b>h

let f be a function such that lim x--> 0, f(2+h) - f(2) /h = 5 , which of the following are true? I. f is continuous at x=2 II. f is differentiable at x=2 III. the derivative of f is continuous at x=2

I and II

which of the following are true about the function x f(x) = ∫ ln(2t-1)dt? 1 I. F(1) =0 II. F'(1)=0 III. F''(1)=1

I and II

the figure above shows the graph f ''(x), the second derivative of a function f(x). the function is continuous for all x. which of the following statements about f are true? I. f is concave up for x<0 and b<x<c II. f has a relative minimum in the open interval b<x<c III. f has points of inflection at x=0 and x=b

I and III

if lim x-->2 f(x)/x-2 = f'(2) = 0, which of the following must be true? I. f(2) = 0 II. f(x) is continuous at x=2 III. f(x) has a horizontal tangent at x=2

I, II, and III

The graph of the function f is shown in the figure above. If the function G is defined by x G(x) = ∫ f(t)dt, for -4≤x≤4 , which of the following -4 statements about G are true? I. G is increasing on (1,2) II. G is decreasing on (-4,3) III. G(0) <0

II and III only

if f is the continuous function show in the figure above, then the area of the shaded region is

a ∫ f(x)dx b

area of a cube

a = 6s^2

let f(x) = x^4 +ax^2 +b. The graph of f has a relative maximum at (0,1) and an inflection point when x=1. The values of a and b are

a=-6 and b=1

the average value of f(x) = e^2x +1 on the interval 0≤x≤0.5 is

e

if f is continuous for a≤x≤b and differentiable for a<x<b, which of the following could be false?

f '(c) = 0 for some c such that a<x<b

let f(x) be a differentiable function with no points of inflection on [a, b]. if the definite integral ∫a to b f(x)dx > t, where T is the trapezoidal rule approximation of ∫a to b f(x) dx, which of the following statements about f(x) must be true?

f(x) is concave downwards on [a, b]

the function f is defined on all the reals such that x^2 +kx-3, if x≤1 f(x) = 3x+b , if x>1 for which of the following values of k and b will the function f be both continuous and differentiable on its entire domain?

k=1 and b=-4

for what nonzero value of k is the line y=x tangent to the curve y=2^x

k=1.884

if y =e^kx, then d^5y/dx^5 =

k^5e^lnx

2 if ∫ (x^2 -x)/x^3 dx = 1

ln2 - 0.5

if c(x) gives you the cost in dollars of producing x items of a certain product, which of the following statement are true about dc/dx, the derivative of c(x)?

the units of dc/dx are dollars per item and the value of dc/dx at any value of x i the cost of producing one additional item

as lim x --> 0 4 (sinxcosx - sinx / x^2 )

undefined

the slope field for the differential equation dy/dx = (x^2y+xy^2)/ (3x+y) will have horizontal segments when

x=0, y=0, or y=-x

∫ln2xdx

xln2x-x+c

let f and g be functions that are differentiable for all real numbers x with g(x) = f(x)/x. if y=2x-3 is an equation of the line tangent to the graph of f at x=1, what is the equation of the line tangent to the graph of g at x=1?

y=3x-4

if g(x) = √x (x-1)^2/3 , then the domain of g' is

{x|0<x<1 or x>1|

if dy/d = (x^3 +1 ) / y and y=2 when x=1, then when x=2, y=

± √(27/2)

the base of a solid is the region enclosed by the ellipse 4x^2 +y^2 = 1. if all plane cross sections perpendicular to the x-axis are semicircles, then its volume is

π/3

∫ from 0 to 1/2 2/(1-x^2)^0.5 dx

π/3

x if sin (3x) - 1 = ∫ f(t)dt, then the value of a is a

π/6