Application of derivatives Questions and Answers

Find the critical points and classify them as local maxima, local minima, saddle points, or none of these. f(x, y) = (x+y)(xy + 9) (x, y)=________ (smaller x-value) (x, y) =_______ (larger x-value)

The revenue from sales of x units of a product is given by R(x) = 200x-0.01x², and the cost of producing and selling the product is C(x) = 38x+0.01x² + 16,000. Producing and selling how many units will result in a profit? Producing and selling between________ and______-units of the products will result in a profit.

(a) If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in terms of (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m/s, how fast is the area of the spill increasing when the radius is 38 m?

A model used for the yield Y of an agricultural crop as a function of the nitrogen level W in the soil (measured in appropriate units) is Y=KN /25 + N² where k is a positive constant. What nitrogen level gives the best yield?

Divide. (-10x³+7x-5)÷(2x²-3) Write your answer in the following form: Quotient +

(-7x4+5x³+20x²+10)÷(-x²+x+3) Write your answer in the following form: Quotient + -7x4+5x³+20x² +10 -x²+x+3 = + 2 -x + x + 3 Remainder 2 -x+x+3 X S ?

Determine if the following function is concave up or concave down in the first quadrant. y=5x^2/3 Is the function y=5x^2/3 concave up or concave down in the first quadrant? Concave up Concave down

Determine whether the function y = -2x³ is increasing or decreasing for the following conditions. (a) x < 0 (b) x > 0 (a) Is the function increasing or decreasing for x < 0? decreasing increasing

The annual total revenue for a product is given by R(x) = 30,000x – 5x² dollars, where x is the number of units sold. To maximize revenue, how many units must be sold? What is the maximum possible annual revenue? To maximize revenue,_________units must be sold. (Simplify your answer.

Let h(x) = -9x - 13 - 6x²- x³ Determine the absolute extrema of h on [-4, 0]. If multiple such values exist, enter the solutions using a comma-separated list. The absolute minimum of h is_______and it occurs at x =_____ The absolute maximum of h is_______and it occurs at x =_____

Consider h(v) = 7v log5( – 6v) on [ – 125/6 ,-1/6] Determine the interval over which h is continuous and the interval over which h is differentiable. h is continuous on _______ h is differentiable on _______ - Use the above information to determine if the Mean Value Theorem may be applied to h over [ – 125/6 ,-1/6]

Find the points on the curve y = x³ + 3x^2 - 9x + 8 where the tangent is horizontal. smaller x-value (x, y) = larger x-value (x, y) =

4 ln(x + 6) x + 6 Let f(x) Determine the absolute extrema of f on [-5, -1]. If multiple such values exist, enter the solutions using a comma-separated list. The absolute minimum of fis +2 The absolute maximum of f is and it occurs at x = and it occurs at x =

Let h(x) - 52 +9 Determine the absolute extrema of h on [-3, 3].

Let g(x) = 17 + 24x + x³ + 9x² Determine the absolute extrema of g on [-5, -1]. If multiple such values exist, enter the solutions using a comma-separated list.

Let h(x) = x² + 21 + 10x Determine the absolute extrema of h on [- 7,2].

For the polynomial below, -3 is a zero. g(x)=x²³ - 2x² - 9x + 18 Express g (x) as a product of linear factors.

7. (Factoring) Factor the difference of squares. a) x² - 121 b) 9m² - 4n²

(1 point) 3sin(x)tan(x)+3¯¯√sin(x)=0 Find all angles in radians that satisfy the equation. For each solution enter first the angle solution in [0,π) оr [0,2π) (depending on the trigonometric function) then the period. When 2 or more solutions are available enter them in increasing order of the angles. (e.g. x=π/2+2kt or x=3π/2+kπ etc.) Note: You are not allowed to use decimals in your answer. Use pi for π.

For the polynomial below, 3 is a zero. g(x)=x²- 4x² + x + 6 Express g (x) as a product of linear factors.

For the polynomial below, -3 is a zero. f(x)=x³ - 3x² Express f(x) as a product of linear factors.

A spherical balloon is inflated at the rate of 67 cm³/sec. At what rate is the radius increasing when r = 4 cm?

Sand falls from an overhead bin and accumulates in a conical pile with a radius that is always two times its height. Suppose the height of the pile increases at a rate of 3 cm/s when the pile is 17 cm high. At what rate is the sand leaving the bin at that instant?

Find the extrema of y = x³-6x² +9x+2 on [0,2]. (Notice this is the same equation as #4a.) Label max/min.

Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) =(t²- 3t, 1 + 4t,1/3 t^3+1/2 t^2) ,t = 3 T(3)=

Find the profit function if cost and revenue are given by C(x) = 178 +4.9x and R(x) = 7x -0.05x². The profit function is P(x) =

Find an equation for the surface consisting of all points that are equidistant from the point (-3, 0, 0) and the plane x = 3. Identify the surface. O parabolic cylinder O hyperbolic paraboloid O hyperboloid of one sheet O circular paraboloid O hyperboloid of two sheets O ellipsoid O elliptic cylinder O cone

Find all local extremes of the function f(x, y) = (x² + y²) e^x²-y²

Find the marginal cost function. C(x) = 190 +3.9x -0.03x²

The total cost (in dollars) of producing x food processors is C(x) = 1900 + 30x -0.1x². (A) Find the exact cost of producing the 91st food processor. (B) Use the marginal cost to approximate the cost of producing the 91st food processor. (A) The exact cost of producing the 91st food processor is $_______ (B) Using the marginal cost, the approximate cost of producing the 91st food processor is $______ *

For f(x)=1/5+x²4the slope of the graph of y = f(x) is known to be -4/81 at the point with x-coordinate 2. Find the equation of the tangent line at that point. _____(Type an equation. Use integers or fractions for any numbers in the equation.).