Calculus Answers

1. Differentiate the following functions:

(a) y = 4x3 + 5x-4 + sin 3x

1.Using a mathematical model and calculus methods (e.g. numerical and

integration methods) to solve given engineering problem (Eq. 1).

v(t) = 3 Cos(πt) − 2Sin(πt) (Eq. 1)

● v(t) is the instantaneous velocity of the car (m/s)

● t is the time in seconds

Your tasks is

a) Derive an equation x (t) for the instantaneous position of the particle as

a function of time using indefinite integration.

b) Sketch a graph of position vs. time over the time interval 0 ≤ t ≤ 3

seconds for Eq.1, where C=12.

c) Find a mathematical model (e.g. equation) to correlate position and

time using an Excel sheet and trendline.

d) Using definite integration and driven equation (from c) to find the area

under curve over the time interval 0 ≤ t ≤ 3 seconds and C=12.

e) Using a mid-ordinate rule and driven equation (from c) to find the area

under curve over the time interval 0 ≤ t ≤ 3 seconds at h= 0.5.

You have been given an equation to calculate velocity of a particle after t (time,

s) and passing a fixed point (P). the equation is:

v(t) = 3 Cos(πt) − 2Sin(πt) (Eq.1)

● v(t) is the instantaneous velocity of the car (m/s)

● t is the time in seconds

Your tasks is:

a) Define the given engineering problem (Eq. 1) and find the

application to use it.

b) Identify what type of calculus methods and mathematical model

can be sued.

c) Analyse the given engineering problem (Eq. 1) and break the

problem down into a series of manageable elements.

d) Explain what type of data and variables can be calculated

(extracted) from a given equation using calculus method and

mathematical method.

e) Explain the reasons for each element (why we used these

methods).

f) How the given engineering problem (Eq.1) can be solved

accurately using calculus methods and mathematical model.

if function f(x,y)={xy/x^2+y^2, (x,y) is not equal to (0,0) 0 , (x,y)=(0,0) is continuous at (0,0).then show that both the partial derivative exists at the origin, but the function is not continuous at the origin.

2+2

5. (Sections 6.1, 6.3) Consider the R − R 2 function r defined by r (t) = ￾ t, t2
; t ∈ [−3, 3] . (a) Determine the vector derivative r 0 (1) by using Definition 6.1.1(b) Sketch the curve r together with the vector r 0 (1), in order to illustrate the geometric meaning of the vector derivative. Note: The curve r is the image of r, so it consists of all points (x, y) = (t, t2 ); t ∈ [−3, 3]

limit as x tends to 2,(root of 6-x, minus 2) all over (root of 3-x, minus 1)

For the function v = 40sin calculate the mean and rms over a range of

1. Use the first principal to differentiate the following with respect to x

f(x) = 3x²- 4x + 7

y=3√x−1/x2+ 1