Practice Problems: Calculus for Physics
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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:
a) x = 2π radians
b) x = 0
c) x = π/8 radians
(Note: Make sure that your calculator is in radians mode.)

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

8. (moderate) Evaluate the indefinite integrals shown below:
a) ∫8e^7xdx
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) ∫e^8x-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) ∫5e^xdx (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