Quiz 8

Names:

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Problem 1: Syntax Corrections (5 × 2 = 10 marks)

The following Mathematica commands have syntax errors. Fix the errors so the commands will work. NOTE: The Clear[] commands below have no syntax errors--they are there to make sure you don't accidentally assign a value in the kernal causing the correct syntax not to work!

Part 1: Plotting

We want to plot (x^2)for -2≤x<2, and we want the plot style to be thick and red.

plot[x^2,(x,-2,2),Style->{Thick,RED}]

Part 2: Solving Equations and Defining Functions

We want to solve the equation f(x)=(Sqrt[4(x^2)-1])/(x+1)=0.

Clear[f]
f[x]=SquareRoot(4x^2-1)/(x+1)
Solve[f[x]=0,x]

Part 3: Solving System of Equations

We want to solve the system of equations 3x+2y=1 and 4x-7y=3.

Clear[x,y]
solve[3x+2y= 1,4x-7y= 3]

Part 4: Differentiation

We want (d/(d x)) ln(x):

Differentiate[ln(x),x]

Part 5: Integration

We want to calculate (∫_0^(2π))(e^(7x))dx:

integrate[e^(7x),{x,0,2pi}]

Problem 2: Basic Mathematica Skills (5 × 2 = 10 marks)

Use Mathematica to carry out the following tasks:

Part 1: Defining Functions

Define the functions f(x)=arccos x and g(x)=(Sqrt[sin x]) and then compute the composition g ◦ f and (g ◦ f)(1/2).

Part 2: Plotting Functions

Plot the parametric function x(t)=(e^(-2t))+cos t, y(t)=sin (3t) for 0<t<10

Part 3: Symbolic and Numeric Calculation

Find x that satisfies a (x^3)+b x+c=0 in terms of the constants a , b, c.

Find x that satisfies 3 (x^3)+2 x-1=0 to 24 decimals.

Part 4: Differentiation

Determine the second derivative A''(t) when A(t)= (1+1/t+ln(2t)) cos t

Part 5: Integration

Compute (∫_0^14)(x/(((x+1))^2))dx exactly, and then determine the value as a decimal.

Bonus: Plotting in 3D (2 marks)

Plot the two functions z=f(x,y)=cos(2x y)-sin(3x y),  and z=g(x,y)=sin( x y) with -π<x<π and -1<y<1 on the same set of axes.