MATH 152000 - Midterm 1

useful hypotehtically

Definition of a lower sum

Suppose f: [a,b] —> R. Suppose P = (x₀, x₁, …. x_n) is a partition of (a, b).

L_f(P) = (n-1) ∑(i=0) of Δx_i * m_i

Definition of an upper sum

Suppose f: [a,b] —> R. Suppose P = (x₀, x₁, …. x_n) is a partition of (a, b).

U_f(P) = (n-1) ∑(i=0) of Δx_i * M_i

d/dx sinx

cosx

d/dx cosx

-sinx

d/dx tanx

sec²x

d/dx secx

-secxtanx

Δx_i

x_i - x_i-1

m_i

minimum of f on [x_i-n, x_i]

M_i

maximum of f on [x_i-n, x_i]

FTC

Suppose f is a continuous function on [a, b]. Let a<c<b. We define F(x) = ∫ (c, x) f(t) dt. Then:

F(x) is differentiable and F’(x) = f(x)

FTC 2

Suppose f is a continuou function on [a, b]. Suppose g is any antiderivative of f. Then:

∫(a, b) f(x) dx = g(b) - g(a)

d/dx cotx

-csc²x

d/dx cscx

-cscxcotx

d/dx ( ∫ (a, u(x)) f(t) dt

= f(u(x)) * u’(x)

When can we say a function f is odd?

f(-x) = -f(x), for all x (ex: sinx)

When can we say a function f is even?

f(-x) = f(x), for all x (ex: cosx)

MVT of integrals

Suppose f is continuous on [a,b]. there exists at least one number c in [a, b] such that 

f(c) = 1/(b-a) * ∫ (a, b) f(x) dx = average of f on [a, b]

∫ (-a, a) f(x) dx when f(x) is ODD

= 0

∫(-a, a) f(x) dx when f(x) is EVEN

2 * ∫(0, a) f(x) dx

MVT for integrals 2

Suppose f and g are continuuous on [a,b]. Assume g is nonnegative. Then there is at least one value c in [a, b] such that:

f(c) = ∫(a, b) f(x) g(x) dx / ∫(a, b) g(x) dx

OR

f(c) * ∫(a, b) g(x) dx = ∫ (a, b) f(x) g(x) dx

Calculating the area of Type 1 Regions

f(x) on top, g(x) on the bottom. x = a and x = b are intersects of f(x) and g(x)

Area = ∫(a, b) (f(x) - g(x)) dx

Calculating the area of Type 2 Regions

f(y) to the right, g(y) to the left. y=a and y = b are intersects of f(y) and g(y)

Area = ∫(a, b) (f(y) - g(y)) dy

Calculating volume of a solid (WASHER Method)

Vol = π  ∫(a, b) ((R(x))² - (r(x))² ) dx

R(x) = outer radius

r(x) = inner radius

Calculating volume of a solid (SHELL Method)

Vol = 2π ∫(a, b) (x * f(x) ) dx

x = radius of shell

f(x) = height of shell

d/dx ln x

1/x

d/dx f^-1 (x)

1 / f’(f^-1(x))

Steps for finding volume of a solid when the base is a function and cross sections are a shape

1. Solve the equation of the area of the base in terms of the axis the cross sections are parallel to (ex. if the cross sections are parallel to the y-axis, solve for y)

2. integrate the area of the base from the points where the region hits the axis that the cross sections are perpendicular to (ex. if the cross sections were parallel to the y-axis, take the bounds from where the equation hits the x-axis)

3. solve the integral! done and dusted, out of this world!

sin(π/6)

1/2

cos(π/6)

√3/2

sin & cos π/4

√2/2

sin π/2

1

cos π/2

0

sin π

0

cos π

-1

cos 0

1

sin 0

0

cos π/3

1/2

sin π/3

√3/2