Calculus Practice
Calculus Practice
In physics, calculus is a tool – it gets you from place A to place B – kind of like your car. Do you understand in detail how your car engine works? Most drivers don’t. They understand their cars operationally – the accelerator makes you go faster, the steering wheel changes your direction, etc. Treat calculus the same way! It is vast, complex and beautiful — and your math course will take you “under the hood”. In the meantime, let’s get you on the road…
Exercise 1: Derivatives
1.1 What is a derivative? What does it tell you?
1.2 Find the derivative of the following:
A. [latex]y(x) = x^3[/latex]
Answer
[latex]\frac{dy}{dx}=3x^2[/latex]
B. [latex]y(x) = 4x[/latex]
Answer
[latex]\frac{dy}{dx}=4[/latex]
C. [latex]y(x) = 0.5x^2[/latex]
Answer
[latex]\frac{dy}{dx}=x[/latex]
D. [latex]y(x) = 10[/latex]
Answer
[latex]\frac{dy}{dx}=0[/latex]
E. [latex]x(t) = t^4[/latex]
Answer
[latex]\frac{dx}{dt}=4t^3[/latex]
F. [latex]x(t) = 4t^3 -6t^2 +2[/latex]
Answer
[latex]\frac{dx}{dt}=12t^2 -12t[/latex]
G. [latex]v(t) = \frac{4}{t^3}[/latex]
Answer
[latex]\frac{dv}{dt}=\frac{-12}{t^4}[/latex]
H. [latex]y(x) = 8x^{1/2}[/latex]
Answer
[latex]\frac{dy}{dx}=4x^{-1/2}[/latex]
I. [latex]y(x) = mx + b[/latex], where [latex]m, b[/latex] are constants.
Answer
[latex]\frac{dy}{dx}=m[/latex]
J. [latex]x(t) = x_0 + v_0t+\frac{1}{2}gt^2[/latex], where [latex]x_0, v_0, g[/latex] are constants.
Answer
[latex]\frac{dx}{dt}=v_0 + gt[/latex]
K. What is the slope of the function: [latex]x(t) = 4t^3 + 3[/latex] at [latex]t=2[/latex]?
Answer
[latex]\frac{dx}{dt} = 12t^2[/latex]
[latex]12t^2[/latex] at [latex]t=2[/latex], [latex]\frac{dx}{dt} = 48.[/latex]
Exercise 2: Anti-derivatives and integrals
2.1 What is an anti-derivative or an integral? What can it tell you?
2.2 Find the original functions:
A. [latex]\frac{dy}{dx} = x^2[/latex]
Answer
[latex]y(x) = \frac{1}{3}x^3 + C[/latex]
B. [latex]\frac{dy}{dx} = 0.2x^3[/latex]
Answer
[latex]0.05x^4 +C[/latex]
C. [latex]\frac{dy}{dx} =4x[/latex]
Answer
[latex]y(x) = 2x^2+C[/latex]
D. [latex]\frac{dy}{dx} = 2[/latex]
Answer
[latex]y(x) = 2x+C[/latex]
E. [latex]\frac{dy}{dx} = 0[/latex]
Answer
[latex]y(x) = C[/latex]
F. [latex]\frac{dx}{dt} = 4t^2 -6t +2[/latex]
Answer
[latex]x(t) = \frac{4}{3}t^3-3t^2 + 2t + C[/latex]
G. [latex]\frac{dv}{dt} = \frac{10}{t^2}[/latex]
Answer
[latex]v(t) = -\frac{10}{t}+C[/latex]
2.3 Determine the integrals:
H. [latex]\int 2x^3\ dx[/latex]
Answer
[latex]= \frac{1}{2}x^4 + C[/latex]
I. [latex]\int x\ dx[/latex]
Answer
[latex]=\frac{1}{2}x^2 + C[/latex]
J. [latex]\int (z^3+5)\ dz[/latex]
Answer
[latex]=\frac{1}{4}z^4 +5z + C[/latex]
K. [latex]\int dx[/latex]
Answer
[latex]=x + C[/latex]
L. [latex]\int (at+v_0)\ dt[/latex], where [latex]a[/latex] and [latex]v_0[/latex] are constants.
Answer
[latex]=\frac{1}{2}at^2 +v_0t + C[/latex]
M. [latex]\int_1^3 0.5t^2\ dt[/latex]
Answer
[latex]=\frac{1}{6}t^3|^{3}_{1}[/latex]
[latex]=13/3[/latex]
N.[latex]\int_{-2}^{2}5t\ dt[/latex]
Answer
[latex]=\frac{5}{2}t^2|_{-2}^{2}[/latex]
[latex]=0[/latex]
O. [latex]\int_{1}^{2}\frac{4}{x^3}\ dx[/latex]
Answer
[latex]=\frac{-2}{x^2}|_{1}^2[/latex]
[latex]=3/2[/latex]