Calculus Answers

Let Z = f(x,y) = 3x3 -5y2 -225x + 70y + 23. (i) Find the stationary points of z. (ii) Determine if at these points the function is at a relative maximum, relative minimum, infixion point, or saddle point. 

The tangents at the origin of the curve x^3+y^3=3axy are

a manufacturer wants to build a spring that takes a force 10 n(in negative direction) to compress it 0.2 m from the equilibrium position.the spring should be able to stretch 0.5 m from the equilibrium position.

as a mechanical engineer, you were asked to present how much work should be done to stretch the spring.the presentationshall include important components of the problems, complete and correct computations and a logical and organizedexplanation.

 Duke’s Lunch Buffet currently sells 2500 lunches per week at a price of $11 per meal. They estimate that for each $1 increase in price, they will lose 150 customers per week. At what price should Duke’s sell their lunches to maximize their weekly revenue?

I. Investigate whether the following functions are even or odd:

(a) f(x) = x³

(b) f(x) = cos x

II. State the mean value theorem

III. (a) Find the derivative of the function y = 2x² + 12/x² when x = 2

(b) f(x) = -3/x-7. Find the inverse of the function.

Iv. Consider the function f(x) = e^rx Determine the values of r so that f satisfies the equation f"(x) + f'(x) - 6f(x) = 0.

Air is escaping from a spherical balloon at the rate of 2𝑐𝑚3 per minute. How fast is the radius shrinking when the volume is 36𝜋 𝑐𝑚3 ?

Find the rate of change of the area 𝐴, of the circle with respect to its circumference C, 𝑖. 𝑒 𝑑𝐴 𝑑�

Let 𝑓(𝑥) = 1 3 𝑥 3 + 𝑥 2 − 15𝑥 − 9 . Use detailed sign tables in answering the following questions. (a) Find the intervals in which 𝑓 is increasing or decreasing. (b) Find the intervals in which the graph of 𝑦 = 𝑓(𝑥) is concave upwards or downwards. 

1. Consider the relation P = {(1,7), (-1,7), (3,9), (1,3). is P a function? Justify your answer.
2. Use sign table to determine the values of x for which x²+ 9x +20 is less than or equals to 0
3. Prove that f(x) = x² + 2x is not injective

Prove that 𝑓(𝑥) = 𝑥 2 + 2𝑥 is not injective

𝑦 = 5𝑒 − 8 Differentiate the given function.