Physics C Mechanics

Practice Problems: Calculus for Physics

1. (easy) Determine the limit for each of the following:
a) lim (x - 8) as x → 4
b) lim (x/2) as x → 10
c) lim (5x + 2) as x→ 3
d) lim (4/x) as x → 0

2. (moderate) Determine the limit for each of the following:
a) lim [(x2 - 6x + 9)/(x - 3)] as x→3
b) lim[(x2 - 3x)/x2] as x→0
c) lim[(x2 - 4)x/x2 as x→2
d) lim(3x3/x4) as x→∞

3. (moderate) Find the derivative (dy/dx) of the following functions with respect to x.
a) y = 2x +5
b) y = 3x2 + 7x
c) y = 5cosx
d) y = 3/x2
e) y = (x + (1/x))(x - (1/x))
f) y = ln(x3)
g) y = 3e-2x
h) y = (6x)sin(2x)

4. (moderate) Find the first, second, and third derivatives of the following functions wth respect to x:
a) y = 9x2 + 3x + 5
b) y = 2/x5
c) y = 10sin(6x)

5. (moderate) Given that y = x3 - 2x, find the slope one would measure on a graph of that function when:
a)
x = 1
b) x = -1
c) x = 2/3

6. (moderate) Given that y = 3cos(4x), find the slope one would measure on a graph of that function when:
b) x = 0

7. (easy) Evaulate the indefinte integrals shown below:
a) ∫4xdx
b) ∫(9x + 6)dx
c) ∫3x2dx
d) ∫(2/x)dx
e) ∫10exdx

8. (moderate) Evaluate the indefinite integrals shown below:
a) ∫8e7xdx
b) ∫(x3 + 6x2 + 4x + 8) dx
c) ∫(2x + 6)(x2 + 6x)2dx (try u-substitution)
d) ∫(4 - x)4dx (try u-substitution)
e) ∫(3x + 5)½ dx (try u-substitution)
f) ∫e8x-6dx (try u-substitution)

9. (easy) Evaluate the definite integrals shown below:
a) ∫5x2dx (from x = 2 to x = 5)
b) ∫12x-1dx (from x = 3 to x = 10)
c) ∫5exdx (from x = -3 to x = -2)
d) ∫20sin(x)dx (from x = 0 to x = π/2) (radians mode on calculator, please)

10. (moderate) Evaluate the definite integrals shown below:
a) ∫xcos(x2 + 3)dx (from x = 4 to x = 6)
b) ∫7cos(6x)dx (from x = 0 to x = π)
(can use a u-substitution is you want)
c) ∫(sin(lnx))dx/x (from x = 10 to x = 15)
d) ∫(x + 4)dx/(x2 + 8x - 7) (from x = 2 to x = 3)

11. (moderate) Find the change in y if x changes from x = 2 to x = 6 for these differential equations:
a) dy/dx = 3x2 + 7
b) dy/dx = 1/x2
c) dy/dx = (x + 1/x)(x - 1/x)

12. (moderate) Find y when x = 3 if y = 1 when x = 2 for:
dy/dx = x3y2