# MATHEMATICAL METHODS, unit 3 - further calculus

1. a^m * a^n = a^m+n2. a^m/a^n = a^m-n3. a^0 = 14. (a^m)^n = a^m*n5. (ab)m = a^m b^m6. (a/b)^m = a^m/b^m7. 1/a^m = a^-m8. n√a = a^1/n9. n√a^m = (n√a)^m = a^m/n

state the index laws

1. y = a^x --> x = loga y2. logm a + logm b = logm a*b3. logm a - logm b = logm a/b4. loga m^n = nloga m5. loga a = 16. loga 1 = 0 7. loga b = logm b / logm a8. a^loga m = m

state the log laws

dy/dx = lim(h->0) f(x+h) - f(x) / h

state the formula for calculating the derivative as a limit using first principles

If f(x) = e^x, thenf'(x) = e^xIf y = e^f(x), thendy/dx = f'(x)e^f(x)

state the derivative of y = e^x and y = e^f(x)

- inverse relationships are the reflection over the line of y = x- the domain and range swap for inverse graphs

state the inverse relationship between y = e^x and y = log e(x)

y = log e(x) dy/dx = 1/xy = log e f(x) dy/dx = f'(x)/f(x)

state the derivative of y = log e(x) and y = log e f(x)

sin(x) = cos(x)sin(f(x)) = cos(f(x)) * f'(x)

state the derivative of sin(x) and sin(f(x))

cos(x) = -sin(x)cos(f(x)) = -sin(f(x)) * f'(x)

state the derivative of cos(x) and cos(f(x))

dy/dx = dy/du * du/dxor aunu^n-1

state the chain rule

vu + uv`

state the product rule

(vu'-uv')/v^2

state the quotient rule

dy/dx (tan(x)) = 1/cos²(x)

state the derivative of tan(x)

∫x^n.dx = 1/n+1 x^(n+1) + c∫(f(x) ± g(x)).dx = ∫f(x).dx ± ∫g(x).dx ∫kf(x).dx = k∫f(x).dx ∫k.dx = k∫1.dx = kx + c

state the general formula for integration (and the three properties)

∫ke^ax.dx = k∫e^ax.dx = k/a e^ax + c

state the formula for antidifferentiation of exponential functions

∫1/x.dx = ln(x) + c∫1/(ax+b).dx = 1/a ln(ax+b) +c

state the formula for antidifferentiation of logarithmic functions

∫sin(ax+b).dx = -1/a cos(ax+b) + c∫cos(ax+b).dx = 1/a sin(ax+b) + c

state the formula for antidifferentiation of sine and cosine functions

∫(ax+b)^n.dx = 1/a(n+1) (ax+b)^n+1 + c

state the chain rule for integration

if d/dx[f(x)] = g(x), then ∫g(x).dx = f(x) + c

state the formula for the basis of integration by recognition

the left end-point rectangle method - slightly less for increasing functions and slightly more for decreasing functionsthe right end-point rectangle method - slightly more for increasing functions and slightly less for decreasing functionsthe trapezoidal method - the average of the left and right end-point areas. using the formula A = w/2 (ends + 2(middles))

state the three methods for estimating the area under a curve

Area = ∫a^b f(x).dx = [F(x)]a^b = F(b)-F(a)

state the formula for the definite integral

-∫a^b f(x).dx∫b^a f(x).dx|∫a^b f(x).dx|therefore, when regions are combined, this is used along with the x-intercepts to ensure the areas don't cancel out

the area between a function and the x-axis using the definite integral can be either positive or negative depending on if the graph is above or below the x-axis. however, since area cannot be negative, the answer is made positive. state the three methods used to do this.

A = ∫a^b [f(x)-g(x)].dxnote that the bottom function is always subtracted from the top function

state the formula for the area between curves

instantaneous rate of change with respect to the number of items (dR/dx)

define the marginal rate of change

the extra revenue received for selling one more unit after a particular number of units have been sold (eg. approximate revenue when 101st unit sold = dR/dx | x=100)

define the marginal revenue

displacement = signed area or definite area (don't change signs)distance = the area under the curve

from a velocity graph, how can the displacement and distance be calculated