Other Functions In Calculus

Natural Log:Differentiation

100

Write the expression as a single logarithm. ln(x-2) - ln(x+2)

ln((x-2)/(x+2))

200

Find the Derivative of the following function: f(x) = ln(2x/(x+3))

f(x) = ln(2x) - ln(x+3) f'(x) = (2/x) - 1/(x+3)

300

Find the derivative for f(x) = ln(e^(x^3))

ln(e) = 1 f(x) = x^3 f'(x) = 3*x^2

400

Use implecit differentiation to find dy/dx for: ln(xy) + 5x = 30

Seperate - ln(x) + ln(y) + 5x = 30 (1/x) + (1/y)y' + 5 = 0 (1/y)y' = -(1/x) - 5 y' = (-y/x) - 5y

500

Use Logarithmic differentiation to find dy/dx for: y=x*(x^2 - 1)^(1/2)

ln(y) = ln(x) + (1/2)*ln(x^2 - 1) (1/y)y' = (1/x) + x/(x^2-1) y' = y[(1/x) + x/(x^2-1)] or y' = y*(2x^2-1)/(x(x^2-1))

Natural Log: Integration

100

Integrate (10/x) dx

10*ln|x| + c

200

Integrate (2x^2 + 7x - 3)/(x-2) dx

Use long division: Integrate (2x + 11 + (19/(x-2)) dx x^2 + 11x + 19*ln|x-2| + c

300

Integrate cos(t)/(1+sin(t)) dt

u = 1+sin(x) du = cos(x) dx ln|1+sin(t)| + c

400

Integrate (1+ln(x))^2 / (x) from 1 to e.

u = 1+ ln(x) (1/3)(1+ln(x))^3 from 1 to e = (8/3) - (1/3) = (7/3)

500

True or False: 1) (ln(x))^.5 = .5*ln(x) 2) S ln(x) dx = (1/x) + c

False 1) .5*ln(x) = ln(x^.5) False 2) dy/dx ln(x) = (1/x) We do not know the integral of ln(x)

Exponential Functions

100

Find dy/dx for y = e^(sqrt(x))

y' = .5*x^(-.5)*e^(sqrt(x))

200

Integrate (e^(2x))/(1+e^(2x))

let u = 1+ e^(2x), then du = 2*e^(2x) .5*the integration of 1/u F(x) = .5* ln(1+e^(2x)) + C

300

Use implicit differentiation to find dy/dx: e^(xy) + x^2 - y^2 = 10

(xy'+y)*e^(xy) + 2*x - 2y*y' = 0 y'(x*e^(xy) - 2y) = -y*e^(xy) - 2*x y' = (-y*e^(xy) - 2*x) / (x*e^(xy) - 2y)

400

Integrate e^(3/x) / (x^2) dx from 1 to 3.

Let u = (3/x) du = (-3/x^2) (-1/3) *e^(3/x) from 1 to 3. (e^3)/3 - e/3

500

Prove (e^a)/(e^b) = e^(a-b) Perhaps ask for a hint.

ln(e^a)/(e^b) = ln(e^a) - ln(e^b) ln(e^a) - ln(e^b) = a - b Also, ln e^(a-b) = a - b a - b = a - b. Q.E.D

Inverse Trig: Differentiation

100

Give the general solution for the following: (NO CALCULATOR!!) arccos(0) = ??

cos(x) = 0 pi/2 + pi*k, where k is an integer.

200

Find dy/dx for: y = arcsin(t^2)

Chain rule! y' = 2*t/(sqrt(1-t^4))

300

True or false (explain why it is so, or give a counter example): arcsin(pi/4) = sqrt(2)/2

False sin(pi/4) = sqrt(2)/2 Thus, arcsin(sqrt(2)/2) = pi/4

400

Find y' for arctan(x) + x/(1+x^2)

y' = 1/(1+x^2) + ((1+x^2)*1 - x*2x))/(1+x^2)^2 y' = ((1+x^2) + (1-x^2))/(1+x^2)^2 y' = 2/ (1+x^2)^2

500

Find any relative extreme for the function (identify it as a maximum or minimum): f(x) = arctan(x) - arctan(x-4)

f'(x) = 1/(1+x^2) - 1/(1+(x-4)^2) = 0 1+x^2 = 1 + (x-4)^2 1+x^2 = 1+x^2 - 8x+16 -8x + 16 =0 x=2 By the first derivative test, (2,2.214) is a relative maximum

Inverse Trig: Integration

100

Integrate f(x) = 1/(4+(x-1)^2)

F(x) = .5*arctan((x-1)/2) + c

200

Integrate 1/(sqrt(4-x^2)) dx from 0 to 1 (show your work)

arcsin(x/2) from 0 to 1 arcsin(.5) - arcsin(0) = pi/6

300

Integrate 1/(x^2 + 4x + 13) Think of the Greeks!

Completing the square for the bottom yields (x+2)^2 -4 + 13 = (x+2)^2 + 9 integrate 1/((x+2)^2 + 9) dx F(x) = (1/3) arctan((x+2)/3) + c

400

Choose one person to answer the following question from memory (or show how it is found) Integrate (-1)/(a^2 + u^2) du

(1/a)*arccot(u/a) + C

500

Inverse Functions (tricked you =] ) Given f(x) = (1/8)*x - 3 and g(x)=x^3 Find (f^(-1) o g^(-1))(1)

f^(-1) = 8(x+3) g^(-1) = x^(1/3) f^(-1)(g^(-1))(1) = f^(-1) (1) f^(-1)(1) = 32