Chapter 1.4, Problem 76ES

Differentiate the following function

A box with a square base and open top must have a volume of 13,500 cm³. Find the dimensions that minimise the amount of material used. Ensure you show your working to demonstrate that it is a minimum.

Consider the equation, f(x) = x*. (a) Using the trapezoidal method with 3 columns, estimate the value of the integral f² f(x)dx. (b) Using the trapezoidal method with 10 columns, estimate the value of the integral f² f(x)dx. You many need software to help you do this (e.g. MATLAB, Excel, Google sheets). (c) Use software to accurately calculate the integral (e.g. Wolfram alpha, Matlab). Using this answer, comment on the answers you found in parts a) and b).

Using the first-principles definition of differentiation, find the derivative of f(x) = = 2x²

Evaluate the following integrals, showing all your working

Differentiate the following function

Question 1. (10 points) A researcher is studying tumours in mice. The growth rate for the volume of the tumour V(t) in cm³ is given by dV = 1.45V(2 In(V+1)). dt (a) (4 pts) Find all the equilibria and determine their stability using the stability condition. (b) (2 pts) Draw the phase plot f(V) versus V where f(V) = V'. You may find it helpful to use Desmos or Wolfram Alpha to plot the graph of f(V) versus V (both are free to use online), or you can plot it by hand if you like. On the plot identify each equilibrium as stable or unstable. (c) (4 pts) Draw direction arrows for the case where the tumour starts at size 3cm³ and for the case where the tumour starts at size 9cm³. Explain in biological terms what happens to the size of each of these tumours at time progresses.

For the system consisting of the two planes:plane 1: -x + y + z = 0plane 2: 3x + y + 3z = 0a) Are the planes parallel and/or coincident? Justify your answer. What does this tell you about the solution to the system?b) Solve the system (if possible). Show a complete solution. If there is a line of intersection express it in parametric form.

Question 2: (10 points) Evaluate the definite integral. Use the following form of the definition of the integral to evaluate the integral: Theorem: Iff is integrable on [a, b], then where Ax = (ba)/n and x₂ = a + i^x. You might need the following formulas. IM³ L² (3x² (3x²+2x- 2x - 1)dx. n [f(z)dz lim f(x)Az a n→∞ i=1 n(n + 1) 2 n i=1 n(n+1)(2n+1) 6