**Practice Problems: Calculus for Physics**Use your notes to help!

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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 [(x^{2} - 6x + 9)/(x - 3)] as x→3

b) lim[(x^{2} - 3x)/x^{2}] as x→0

c) lim[(x^{2 }- 4)x/x^{2} as x→2

d) lim(3x^{3}/x^{4}) as x→∞

3. (moderate) Find the derivative (dy/dx) of the following functions with respect to x.

a) y = 2x +5

b) y = 3x^{2} + 7x

c) y = 5cosx

d) y = 3/x^{2}

e) y = (x + (1/x))(x - (1/x))

f) y = ln(x^{3})

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 = 9x^{2} + 3x + 5

b) y = 2/x^{5}

c) y = 10sin(6x)

5. (moderate) Given that y = x^{3} - 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 indefinte integrals shown below:

a) ∫4xdx

b) ∫(9x + 6)dx

c) ∫3x^{2}dx

d) ∫(2/x)dx

e) ∫10e^{x}dx

8. (moderate) Evaluate the indefinite integrals shown below:

a) ∫8e^{7x}dx

b) ∫(x^{3} + 6x^{2} + 4x + 8) dx

c) ∫(2x + 6)(x^{2} + 6x)^{2}dx (try u-substitution)

d) ∫(4 - x)^{4}dx (try u-substitution)

e) ∫(3x + 5)^{½} dx (try u-substitution)

f) ∫e^{8x-6}dx (try u-substitution)

9. (easy) Evaluate the definite integrals shown below:

a) ∫5x^{2}dx (from x = 2 to x = 5)

b) ∫12x^{-1}dx (from x = 3 to x = 10)

c) ∫5e^{x}dx (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(x^{2} + 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/(x^{2} + 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 = 3x^{2} + 7

b) dy/dx = 1/x^{2}

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 = x^{3}y^{2}