Math 1B Midterm 1

cos(x)

derivative of sin(x)

-sin(x)

derivative of cos(x)

sec²(x)

derivative of tan(x)

-csc²(x)

derivative of cot(x)

sec(x)tan(x)

derivative of sec(x)

-csc(x)cot(x)

derivative of csc(x)

e^x

derivative of e^x

a^x ln a

derivative of a^x

1/x

derivative of ln(x)

1/(x ln b)

derivative of log_b(x)

1/(√(1-x²))

derivative of sin⁻¹(x)

-1/(√(1-x²))

derivative of cos⁻¹(x)

1/(1+x²)

derivative of tan⁻¹(x)

-cos(x) + C

integral of sin(x)

sin(x) + C

integral of cos(x)

-ln|cos x| + C

integral of tan(x)

ln|sin x| + C

integral of cot(x)

ln|sec x + tan x| + C

integral of sec(x)

-ln |cscx + cotx| + C

integral of csc(x)

sin²x + cos²x = 1

pythagorean identity #1

tan²x + 1 = sec²x

pythagorean identity #2

sin(a)cos(b) + cos(a)sin(b)

sin(a + b) =

sin(a)cos(b) - cos(a)sin(b)

sin(a - b) =

cos(a)cos(b) - sin(a)sin(b)

cos(a + b) =

cos(a)cos(b) + sin(a)sin(b)

cos(a - b) =

(tan(a) + tan(b)) / (1 - tan(a)tan(b))

tan(a + b) =

(tan(a) - tan(b)) / (1 + tan(a)tan(b))

tan(a - b) =

2sin(x)cos(x)

sin(2x) =

cos²(x) - sin²(x) 2cos²(x) - 1 1 - 2sin²(x)

cos(2x) =

2tan(x)/ (1-tan²(x))

tan(2x) =

(1-cos(2x)/2

sin²(x) =

(1+cos(2x)/2

cos²(x) =

Substitute x = asinθ √a²-x² → acosθ

For √a²-x²

Substitute x = atanθ √a²-x² → asecθ

For √a²+x²

Substitute x = asecθ √x²-a² → atanθ

For √x²-a²

(b-a)/n

Δx

Mn = Δx [f(x̄1) + f(x̄2) + ... + f(x̄n)]

Midpoint Approximation

Tn = Δx/2 [f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)]

Trapezoidal Rule

A ≈ ∆x/3 [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + f(xn)]

Simpson's Rule

|ET| ≤ k(b-a)³/12n²

Trapezoidal Rule Error Bound

|EM| ≤ k(b-a)³/24n²

Midpoint Approximation Error Bound

|ES| ≤ k(b-a)⁵/180n⁴

Simpson's Rule Error Bound

L = from a to b ∫√(1 + [f'(x)]²) dx

Length Formula

S = from a to b 2π∫f(x)√(1 + [f'(x)]²)dx

Surface Area x-axis Formula

S = from a to b 2π∫x√(1 + [f'(x)]²)dx

Surface Area y-axis Formula

x̄ = from a to b (1/A)∫x[f(x) - g(x)]dx

Center of Mass for x̄

ȳ = from a to b (1/A)∫(1/2)[f(x)² - g(x)²]dx

Center of Mass for ȳ

From A to 0 ∫[p(x) - B] dx

Consumer Surplus (CS)

From A to 0 ∫[B - ps(x)] dx

Producer Surplus (PS)