Calculus Answers

3. Suppose that f and g are continuous on [a, b] and differentiable on (a, b). Suppose also that f(a) = g(a) and f 0 (x) < g0 (x) for a < x < b. Prove that f(b) < g(b)

(b) By considering the derivative of the function f : [−1, 1] → R defined by f(x) = \frac{2x}{\:x^2+1} , show that f^{-1} exists and find \left(f^{-1}\right)^' ( \frac{4}{\:5} ).

2. (a) Express 5 sinh x + cosh x in the form Ae^x+Be^{-x}, where A and B are integers.

(b) Solve the equation 5 sinh x + cosh x + 5 = 0, giving your answer in the form ln a, where a ∈ R.

Find the domain and range of the function f, defined by f(x,y) = 2xy/( x²+y ²) . Also find two level curves of this function. Give a rough sketch of them.

Find the angle between the curves y^2=ax and ay^2=x^3 (a>0), at the points of intersection other than origin.

The supply function of a certain product is P = 16 + 2Q, where P is the price and Q is the number of units produced. Find the producer surplus if the market price P = 200. [1] 7 400 [2] 8 464 [3] 12 464 [4] 20 000

If the demand function of a commodity is Q = 36 − 4P, where P and Q are price and quantity respectively, determine the price elasticity of demand when the price is R5. Indicate whether demand is elastic or inelastic at this price and provide justification for your answer. [1] εd = −1,25; because | − 1,25| = 1,25 > 1, demand is elastic [2] εd = −0,25; because | − 0,25| = −0,25 < 1, demand is elastic [3] εd = 0,25; because |0,25| = 0,25 < 1, demand is inelastic [4] εd = 1,25; because |1,25| = 1,25 > 1, demand is inelastic 

Find the total work done in moving a particle in a force field given by 𝐹 = 3𝑥𝑦𝑖 − 5𝑧𝑗 + 10𝑥𝑘 along the curve x = 𝑡² + 1, 𝑦 = 2𝑡² , 𝑧 = 𝑡³ 𝑓𝑟𝑜𝑚 𝑡 = 1 𝑡𝑜 𝑡 = 2.

Find the total work done in moving a particle in a force field given by 𝐹 = 3𝑥𝑦𝑖 − 5𝑧𝑗 + 10𝑥𝑘 along the curve x = 𝑡 2 + 1, 𝑦 = 2𝑡2 , 𝑧 = 𝑡 3 𝑓𝑟𝑜𝑚 𝑡 = 1 𝑡𝑜 𝑡 = 2.

 Verify Stokes’ theorem for 𝐴 = 2𝑥 − 𝑦 𝑖 − 𝑦𝑧² 𝑗 − 𝑦²𝑧𝑘, where S is the upper half of the sphere 𝑥²+ 𝑦² + 𝑧² = 1 and C is its boundary.