Limits
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).
f(x) = sqr(x) + 9*3root(x^7) - 2/(5root(x^2))
x^-1/2 + 21x^4/3 + 4/5x^-7/5
f(x) = 4^x - 5log9(x)
4^x ln(4) - 5/(xln9)
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.
What points do you have to test to find the absolute/local minima?
Critical points (when derivative is 0, DNE), end points
lim x -> 2 (x^2 + 4x - 12)/(x^2 - 2x)
4
f(x) = (3x + 9)/(2 - x)
15/(2 - x)^2
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)
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
Determine the absolute extrema for f(x) = 2x^3 + 3t^2 - 12t + 4 on [-4,2]
max @ (-2,24)
min @ (-4,-28)
lim x -> 0 (cos(4x) - 1)/x
0
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
f(x) = 5e^x / (3e^x + 1)
5e^x/(3e^x + 1)^2
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
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
lim x -> 0 ((1/x+1)-1)/x
-1
f(x) = ln(sin(x)) - (x^4 - 3x)^10
cot(x) - 10(4x^3 - 3)(x^4 - 3x)^9
f(x) = sin-1(x)/(1 + x)
((1/x)/(sqr(1-x^2)) - sin-1(x))/(1+x)^2
Determine all the numbers c which satisfy the conclusions of the Rolles Theorem for f(x) =- -3x*sqr(x + 1).
-2/3
Determine the possible inflection points for f(x) = e^(4-x^2)
x = +- 1/(sqr(2))
lim x -> 0 x/(3 - sqr(x+9))
-6
f(x) = (1 + e^-2x)/(x + tan(12x))
[-2e^-2x(x + tan12x) - (1 + e^-2x)(1 + 12sec^2(12x))]/(x + tan12x)^2
4x^2y^7 - 2x = x^5 + 4y^3
(8xy^7 - 5x^4 - 2)/(12y^2 - 28x^2 y^6)
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
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