1325 Business Calc Final (Quiz Qs) HCC, Ch 11.1
Suppose that the total profit in hundreds of dollars from selling x items is given by P(x)= -x^2 +12x-33. Find the marginal profit at x = 4.
$400 per item
The revenue generated by the sale of x bicycles is given by R(x)= 70.00x-(x^2/200) Find the marginal revenue when x = 1300 units
$57.00/unit
Find the absolute extremum within the specified domain. Maximum of f(x) = (x + 1)2(x - 2); [-2, 1]
(-1, 0)
Find the point from those given that has the given property. The point where the slope of the tangent is greatest
(-1,1) negative parabola
Write interval notation for the graph.
(-3, 1]
Find the largest open interval where the function is changing as requested. Decreasing f(x) = √(4-x)
(-∞, 4)
The rule of the derivative of a function f is given. Find the location of all points of inflection of the function f. f'(x) = (x + 4)(x + 1)(x - 5)
(0 - √63)/3, (0 + √63)/3
Find the coordinates of the points of inflection for the function. f(x) = 4x/(x^2 + 9)
(0,0), (√27, 1/9√27), (-√27, -1/9√27)
Find the largest open interval where the function is changing as requested. Increasing f(x) = x2 - 2x + 1
(1, ∞)
Find the largest open intervals where the function is concave upward. f(x) = x^3 - 3x^2 - 4x + 5
(1, ∞)
The point where the slope of the tangent is least. [Hint: Identify the point that can produce the tangent line with highest slope]
(1,3) sqwiggle down left and up right
Find the absolute extremum within the specified domain. Minimum of f(x) = x3 - 3x2; [- 0.5, 4]
(2, -4)
Find dy/dx by implicit differentiation. x1/3 - y1/3 = 1
(y/x)^2/3
Find dy/dx by implicit differentiation. xy + x = 2
-(1 + y)/x
Find dy/dx at the given point. 4xe7y = 19; (1, 0)
-1/7
y = 6x^-2 + 6x^3 + 13x
-12x^-3 + 18x^2 + 13
A ball is thrown vertically upward from the ground at a velocity of 144 feet per second. Its distance from the ground after t seconds is given by s(t) = -16t^2+144t. How fast is the ball moving 5 seconds after being thrown?
-16 ft per sec
y = x^3 + x^2 - 8x - 7 between x = 0 and x = 2
-2
The total profit (in hundreds of dollars) from selling x items is given by P(x)= 8x-5 / 4x+5. Find the marginal average profit function.
-32x^2+40x+25 / (4x^2+5x)^2
Suppose the demand for a certain item is given by D(p) = -4p^2+2p+2 where p represents the price of the item. Find D'(7).
-54
Find dy/dx at the given point. 7x ln y = 16; (1, e)
-e
A voltage source of 12 volts can supply a voltage V (in volts) through a 72-ohm line to a variable resistance R (in ohms) given by the following equation. V= 12R/72+R How fast does V change with respect to R when R is 61 ohms?
0.049 V/ohms
The body-mass index (BMI) is calculated using the equation BMI = (703w)/h^2, where w is in pounds and h is in inches. Find the rate of change of BMI with respect to weight for Sally, who is 64" tall and weighs 120 lbs. If both Sally and her brother Jesse gain the same small amount of weight, who will see the largest increase in BMI? Jesse is 70" tall and weighs 190 lbs.
0.172, Sally
Solve each problem. The velocity of a particle (in ft/s) is given by v = t^2 - 2t +6, where t is the time (in seconds) for which it has traveled. Find the time at which the velocity is at a minimum.
1 s
f(x) = ln x/e^x
1-xln x / xe^x
Suppose that the population of a certain type of insect in a region near the equator is given by P(t) = 22 ln (t+8), where t represents the time in days. Find the rate of change of the population when t=4.
1.8 insects/day
y = (2x-1)^1/2 between x = 1 and x = 5
1/2
f(X) = 4th root x (x>0)
1/4x^-3/4
y= ln |-9x|
1/x
y= ln (x-7)
1/x-7
The position of a particle moving on a number line at time t is given by s(t). Find the instantaneous velocity at time t = 0 and t = 2. s(t)= 3t^2-2t-1
10
A balloon used in surgical procedures is cylindrical in shape. As it expands outward, assume that the length remains a constant 60.0 mm. Find the rate of change of surface area with respect to radius when the radius is 0.080 mm. (Answer can be left in terms of π).
120.32 pi mm^2/mm
Let f(x) = 8x^2 - 5x and g(x) = 7x + 9. Find the composite. f[g(-3)]
1212
f'(4) if f(x) = 9x^5/2 - 7x^3/2
159
x^3-6 / (3x+3)(5x+6)
15x^4+66x^3+54x^2+180x+198 / (3x+3)^2(5x+6)^2
A balloon used in surgical procedures is cylindrical in shape. As it expands outward, assume that the length remains a constant 90.0 mm. Find the rate of change of surface area with respect to radius when the radius is 0.070 mm. (Answer can be left in terms of π). Surface area of cylinder.
180.28π mm2/mm
The power P (in W) generated by a particular windmill is given by P=0.015V^3, where V is the velocity of the wind (in mph). Estimate the instantaneous rate of change of power with respect to velocity when the velocity is 7.4 mph. (Round to one decimal.)
2.5 W/mph
20 to 50
2/3
The total profit from selling x units of cookbooks is P(x) = (4x-7)(7x-5). Find the marginal average profit function.
28-35/x^2
y = x^2 lnx^2
2x(1+lnx^2)
Solve the problem. A company wishes to manufacture a box with a volume of 40 cubic feet that is open on top and is twice as long as it is wide. Find the width to the nearest foot of the box that can be produced using the minimum amount of material.
3 ft
The power P (in W) generated by a particular windmill is given by P=0.015 V^3 where V is the velocity of the wind (in mph). Find the instantaneous rate of change of power with respect to velocity when the velocity is 8.2 mph.
3.0 W/mph
Compute the instantaneous rate of change of the function at at x = a. f(x) = -8x^2-x, a=-2.
31
y = 3e^x / 2e^x+1
3e^x/(2e^x+1)^2
Solve the problem. A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of 43 ft^3. What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.
4.4 ft by 4.4 ft by 2.2 ft
Solve the problem. A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of 53 ft^3. What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.
4.7 ft by 4.7 ft by 2.4 ft
f(x)= ((x^2+1)/(x+5))(x^3-1)
4x^5+25x^4+2x^3+14x^2-10x+1 / (x+5)^2
For a motorcycle traveling at speed v (in mph) when the brakes are applied, the distance d (in feet) required to stop the motorcycle may be approximated by the formula d = 0.05 v2 + v. Find the instantaneous rate of change of distance with respect to velocity when the speed is 49 mph. [Hint: d is a function v]
5.9 mph
Of the given values of x, identify those at which the function is continuous. f(x) = (x+6) / (x+6)(x-5) ; x= 6,0,-6,5
6, 0
Solve each problem. The price P of a certain computer system decreases immediately after its introduction and then increases. If the price P is estimated by the formula P = 160t^2 - 2100t + 6000, where t is the time in months from its introduction, find the time until the minimum price is reached.
6.6 months
Find dy/dx at the given point. 2xy - 2x + y = -14; (2, -2)
6/5
Solve the problem. Find the acceleration function a(t) if s(f) = 2/(4t - 3).
64/(4t-3)^3
y = ln (2x^3-x^2)
6x-2/2x^2-x
y = 4e^x^2
8xe^x^2
Assume x and y are functions of t. Evaluate dy/dt. xy + x = 12; dx/dt = -3, x = 2, y = 5
9
Decide whether or not the function is continuous in the indicated x-interval. -3 to 0
Continous
A particle has a position function s(t). Is the derivative of s(t) with respect to time at a given time t1, the average or instantaneous velocity?
Instantaneous velocity
At the current production level the marginal cost of a company is zero; there is a positive marginal profit at a slightly lower production level; and there is a negative marginal profit at a slightly higher production level. Should the production level be adjusted, and if so, in what direction?
It should not be adjusted.
Determine the location of each local extremum of the function. f(x) = -x3- 7.5x2 - 12x + 3
Local maximum at -1; local minimum at -4
Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. f(x) = x + 2/x
Local maximum at -√2; local minimum at √2
Use the first derivative test to determine the location of each local extremum and the value of the function at that extremum. f(x) =(x^2)/(x^2 + 2)
Local minimum at (0, 0)
Determine the location of each local extremum of the function. f(x) = x3- 6x2 + 12x - 2
No local extrema
Decide whether or not the function is continuous in the indicated x-interval. -4 to 2
Not continuous
The total revenue from the sale of x stereos is given by R(x) = 4000(1-(x/400))^2 Find the marginal average revenue.
R'(x) = 0.025 - (4000/x^2)
is continuous at a. What value(s) of a allow(s) this statement to be true? Give a complete list of the values a could have.
The function would be continuous if a were either 0 or 1.
Find the coordinates of the points of inflection for the function. f(x) = √(x + 9)
There are no points of inflection.
x^2, x < 1 f(x) = { ax+b, 1 < x < 3 x+6, x > 3
a = 4, b = -3
Solve the problem. The demand equation for a certain product is 5p2 + q2 = 1100, where p is the price per unit in dollars and q is the number of units demanded. Find dp/dq.
dp/dq = -q/5p
Solve the problem. At a certain copy center, the cost y (in dollars) of purchasing x thousand printed cards can be approximated by xy + y = 120x. Use implicit differentiation to find and interpret dy/dx when x = 1 and y = 60.
dy/dx = 30 means that ordering an additional thousand cards will cost approximately $30.00 more.
y=e^ex
e^ex+1
f(x) = e^x ln x, (x>0)
e^x(x ln x+1) / x
Evaluate f"(c) at the point. f(x) = (x^2 + 2)/(3x^2 - 1), c = 0
f"(0) = -14
Evaluate f"(c) at the point. f(x) = e^(3x^2 - 3), c = 1
f"(1) = 42
f(x) = (x^2 - 2x + 2)(5x^3 - x^2 + 5)
f'(x) = 25x^4 - 44x^3 + 36x^2 + 6x - 10
f(x) = x^2 + 7x - 2, f'(0)
f'(x) = 2x + 7; f'(0) = 7
f(x) = (6x-2)(5x+7)^2
f'(x) = 6(5x+7)^2+10(6x-2)(5x+7)
f(x) = (x)^1/2(4x-3)+16x-12
f'(x) = 6x^1/2 - 1.5x^-1/2 + 16
f(x) = (x^3-8)^2/3
f'(x)= (2x^2)/(x^3-8)^1/3
f(x) = (4x-3)((x)^1/2+2)
f'(x)= 6x^1/2-1.5x^-1/2+8
f(x) = (3x^4 + 8)2
f'(x)= 72x^7+192x^3
f(x)= 1/(x-5) ; g(x) = x+5
f[g(X)] = 1/x g[f(X)] = (5x-24)/(x-5)
f(x) = 2/x ; g(x) = 2x^3
f[g(x)]= 1/x^3; g[f(X)] = 16/x^3
g(x) = x^2+5/(x+6)^2
g'(x) = 12x-10/(x+6)^3
g(x) = -2/x , g'(-2)
g'(x) = 2/x^2; g'(-2) = 1/2
f(x) = 1/(5x+2)^1/2
g(X) = x^-1/2, h(x) = 5x+2
s is the distance (in ft) traveled in time t (in s) by a particle. Find the velocity and acceleration at the given time. s = √(t^2 - 5), t = 3
v = 1.5 ft/s, a = -5/8 ft/s2
Find all points where the function is discontinuous.
x = -2, x = 0, x = 2
The rate that a moving company charges depends on the distance, x, in miles that the articles are moved. Let C(x) represent the cost to perform a move of x miles. One firm charges as follows:
x = 250 miles
Find the equation of the tangent line at the given point on the curve. x2 + 3y2 = 13; (1, 2)
y = -1/6x + 13/6
Find the equation of the tangent line to the curve when x has the given value. f(x) = 3x^2 + 5x - 7 ; x = -2
y = -7x - 19
Find the equation of the tangent line at the given point on the curve. 2xy - 2x + y = -14; (2, -2)
y = 6/5x - 22/5
y = x^2+2x-2 / x^2-2x+2
y' = -4x^2+8x / (x^2-2x+2)^2
y= x^2-3x+2 / x^7-2
y' = -5x^8+18x^7-14x^6-4x+6 / (x^7-2)^2
y = (3x^2 + 5x + 1)^3/2
y' = 3/2(6x+5)(3x^2+5x+1)^1/2
y= x^2+8x+3 / (x)^1/2
y' = 3x^2+8x-3 / 2x^3/2
y= x^3 / (x-1)^3
y'= -3x^2/(x-1)^4
Find dy/dx by implicit differentiation. 2xy - y2 = 1
y/(y-x)
Find the equation of the tangent line to the curve when x has the given value. f(x) = (x)^1/2 ; x = 4
y= (x/4)+1
Find the equation of the tangent line to the curve when x has the given value. f(x) = x^2 + 11x - 15 ; x = 1
y= 13x-16
Find the equation of the tangent line to the curve when x has the given value. f(x) = (x)^1/2 ; x = 4
y= x/4 +1
