Calculus Answers

Investigate the continuity of the fumction
f(x) = { 3- x, x is geeater than and equal to 2 }
{ x+1, x> 2 }

If it is discontinuous, identify the type of discontinuity. Sketch the graph.

Question 1

(i) If (x + 3) is a factor of the
expression x^3 + 3x^2 - x - 3, find the other two factors?

(ii) If the expression above was divided by (x + 2), what is the remainder?

Question 2

(a) Sketch the graphs on the same axis.
(i) y = x^2 - 4
(ii) y = -x + 3

(b) What are the inequalities of the region bounded (enclosed area) by the two graphs.

Find the maximum value of the function f(x) = x + x2

the net of the tank being cut from a single piece of steel 240cm x 190 cm and can be made using portrait (widthwise) or landscape (lengthwise) orientation. the tank is a cylinder
By examining the volume including all of the constraints, find which way to cut out the net of the tank to find the maximum volume.
radius = x

A person decides to convert their car from petrol to LPG by installing an LPG tank in their boot. In order for the change to be viable the size of the tank needs to be maximised. The tank must be cylindrical in shape, with the net of the tank being cut from a single piece of steel 240cm x 190 cm and can be made using portrait (widthwise) or landscape (lengthwise) orientation. Note: the ends will be circular and cannot be completely detached from the rectangle.

A valid stationary point may not be found due to certain constraints placed on a situation (greatest and least values). By examining the volume including all of the constraints, find which way to cut out the net of the tank to find the maximum volume.

State any assumptions required to generate the solution and give strengths and limitations of the model considering that the car is medium sized.

A person decides to convert their car from petrol to LPG by installing an LPG tank in their boot. In order for the change to be viable the size of the tank needs to be maximised. The tank must be cylindrical in shape, with the net of the tank being cut from a single piece of steel 240cm x 190 cm and can be made using portrait (widthwise) or landscape (lengthwise) orientation. Note: the ends will be circular and cannot be completely detached from the rectangle.

A valid stationary point may not be found due to certain constraints placed on a situation (greatest and least values). By examining the volume including all of the constraints, find which way to cut out the net of the tank to find the maximum volume.

State any assumptions required to generate the solution and give strengths and limitations of the model considering that the car is medium sized.

Generate a general formula, in simplest form, that will give the maximum volume of a rectangular based prism made from a rectangular piece of paper. The dimensions of the rectangular page are w ×2w with a square of dimensions x×x cut from each corner.

Differentiate tan^-1((sins-Cosx)/(Sini Cox’s))

The following 3 points are on a parabola defining the edge of a ski.
(-4,1), (-2,0.94),(0,1)
The general form for the equation of a parabola is
Ax^2+Bx+C=y

1.Use the x- and y-values of 1 of the points to build a linear equation with 3 variables: A, B, and C. Record your equation here.
2.Repeat this process with 1 of the other 2 points to build a 2nd linear equation. Record your equation here.
3.Repeat this process with the other point to build a 3rd equation. Record your equation here.
4.Build a matrix equation that represents this system of equations. Record your matrix equation here.
5.Use a graphing calculator or other graphing utility to find the inverse of the coefficient matrix. Record your result here.
6.Use the inverse matrix to solve the system of equations. Record the equation of the parabola here.

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The following chart shows Peter's monthly electricity usage (in kWh).
Find a function to model the data. (Estimate the values given in the chart to a multiple of 25, and set January at t=0)
E(t)=
The link for this graph
https://api.agilixbuzz.com/Resz/~0.74QNeMFvrichWLKH.PJXjwGodOZLtToelsbWjnqGHv9nuADk3w1AE8HXr22o/57829733,B7B,7/Assets/Media/Images/2c-Bank-Chart-01e.jpg