Limits
Derivatives
Other Derivatives
Applications of Derivatives
Graphing
100
What determines if a limit exists and how is it different from continuity?

If the right and left limit are equal to each other. Continuity also requires the value of f(a) to be equal to the value of lim x->a f(x).

100

f(x) = sqr(x) + 9*3root(x^7) - 2/(5root(x^2))

x^-1/2 + 21x^4/3 + 4/5x^-7/5

100

f(x) = 4^x - 5log9(x)

4^x ln(4) - 5/(xln9)

100

Describe exactly how physics and derivatives relate and how to determine when an object is slowing down or speeding up.

s'(t) = v(t), v'(t) = a(t)

when a(t) and v(t) have opposite signs then slow down, vice versa.

100

What points do you have to test to find the absolute/local minima?

Critical points (when derivative is 0, DNE), end points

200

lim x -> 2 (x^2 + 4x - 12)/(x^2 - 2x)

4

200

f(x) = (3x + 9)/(2 - x)

15/(2 - x)^2

200

Find the equation of the tangent line to x^2 + y^2 = 9 at the point (2, sqr(5))

y = sqr(5) - [2/(sqr(5)](x - 2)

200

The position of an object at any time t is given by s(t) = (t + 1)/(t + 4).

a) determine velocity at any time t 

b) does the object ever stop moving? when?

a) 3/(t + 4)^2

b) never stops moving

200

Determine the absolute extrema for f(x) = 2x^3 + 3t^2 - 12t + 4 on [-4,2]

max @ (-2,24)

min @ (-4,-28)

300

lim x -> 0 (cos(4x) - 1)/x

0

300

f(x) = x^6(sqr(6x^2 - x))

6x^5(5x^2 - x)^1/2 + 1/2x^6(10x - 1)(5x^2 - x)^-1/2

300

f(x) = 5e^x / (3e^x + 1)

5e^x/(3e^x + 1)^2

300

Determine the linear approximation for f(x)=3√x at x=8. Use the linear approximation to approximate the value of 3√8.05

L(x) = 2 + 1/12(x - 8)

L(8.05) ~ 2

300
For f(x) = 2x^3 - 9x^2 - 60x

a) Determine the intervals on which the function increases and decreases

b) Determine the relative max/mins

a) increasing (-inf,-2) U (5,inf), decreasing (-2,5)

b) x = -2 is a relative max, x = 5 is a relative min

400

lim x -> 0 ((1/x+1)-1)/x

-1

400

f(x) = ln(sin(x)) - (x^4 - 3x)^10

cot(x) - 10(4x^3 - 3)(x^4 - 3x)^9

400

f(x) = sin-1(x)/(1 + x)

((1/x)/(sqr(1-x^2)) - sin-1(x))/(1+x)^2

400

Determine all the numbers c which satisfy the conclusions of the Rolles Theorem for f(x) =- -3x*sqr(x + 1).

-2/3

400

Determine the possible inflection points for f(x) = e^(4-x^2)

x = +- 1/(sqr(2))

500

lim x -> 0 x/(3 - sqr(x+9))

-6

500

f(x) = (1 + e^-2x)/(x + tan(12x))

[-2e^-2x(x + tan12x) - (1 + e^-2x)(1 + 12sec^2(12x))]/(x + tan12x)^2

500

4x^2y^7 - 2x = x^5 + 4y^3

(8xy^7 - 5x^4 - 2)/(12y^2 - 28x^2 y^6)

500

Determine all the numbers c which satisfy the conclusions of the MVT for f(x) = x^3 + 2x^2 - x on [-1,2]

[-4 + sqr(76)]/6

500

for f(x) = x^5 - 5x^4 + 8 answer the following:

a) identify critical points

b) determine intervals on which the function increases and decreases

c) classify the critical points as relative max/mins

d) determine the intervals where the function is concave up/down

e) determine inflection points

f) sketch function

a) t = 0,4

b) inc (-inf,0)U(4,inf), dec (0,4)

c) relative min @ t = 0, relative max @ t = 4

d) concave up (3, inf), concave down (-inf,0)U(0,3)

e) t = 3