1
Find the derivative :
1. ax 2. eg(x)
1. ln a (ax)
2. g' (x) eg(x)
What is the limit if 2x2 / (x+5) as x approaches -5 ?
DNE
f(x) = 3x+ x2 Find ∫20 f(x) dx
26 / 3
d/dx (sec-1 x)
1/ x(x2-1)1/2
Take the Anti-derivatives :
∫ sec2 (x) dx
tan (x) +C
Let L(x) = g (f(2x)), find L' (2)
24
If P(t) = 2 sin t, then find P19 (2π/3)
1
For a differentiable function g(x), it is known that g(2) = 7, g'(2) =-3, g''(2) = -5.
Use the tangent line approximation at x=2 to estimate g(2.1).
6.7
Given f(x) =x2 on the interval [-2,1], and find the values of c in the open interval (-2,1). (MVT problem)
c= -1/2
limx-1 ln(x) / x2 - 1
L' Hospital 's Rule!
ans : 1/2
f'(x)=(4-x)x-3 for x >0
Find all intervals on which the graph of f is concave down?
f concave down on 0<x<6.
Find the local linear approximation of f(x) = x3-2x+3 at the point where x=2. Use your approximation to estimate f(2.1), f(1.9), f(1.99).
The local linear approximation: y=10(x-2)+7
f(2.1)=8, f(1.9)=6, f(1.99)=6.9
Suppose that f is an integrable function and that
∫01 f(X) dx =2, ∫02 f(x) dx=1, ∫24 f(x) dx=7
Find ∫04 f(x) dx
8
Evaluate definite integral by using the Net Change Theorem.
∫40 |x-3| dx
5
f' (x) = (4-x)x-3
Find the x-coordinate for the critical point of f. Determine whether the point is a relative max, a relative min.
Critical point: x=4
f has a max at x=4, because f'(x) changes from + to -.
(Calculator)
A sugar ant crawls along the vertical edge of a cereal box with a velocity given by v(t)=2-t +(t-2)2 +(t-2)3-(t-2)4 +2, for [0,3].
For what intervals is the speed of the sugar ant decreasing?
(0,0.776) and (1.474, 2.103)
Let f(x) = 1/3 x3 +x2 -48x +5.
For which x-value(s) does f(x) have a relative maximum?
x=-8
Determine limh→0 (1/(x+h) - 1/x) / h
-1/x2
Given f(x) = (-18x3 +45)3/2 find the equation of the tangent line at x=1 in the form y=mx+b, Use the tangent line to approximate f (1.1)
m=-2
Equation : y=-2x+5
L(1.1) = -2(1.1) +5 = 2.8
Find two positive numbers whose product is 220 and whose sum is as small as possible?
Primary Equation: S=x+y Domain: x>0, y>0
Secondary Equation: xy=220, y= 220/x
S=x+220/x =
If gas is pumped into a spherical balloon at the rate of 5 ft3 / min, at what rate is the radius increasing when r= 3 ft.
dv/dt = 5 = 4πr2 dr/dt, V=4/3 πr3
Find dr/dt
5=4π(3)2 dr/dt
dr/dt = 5/36π ft/min
(Calculator)
Let R be the region bounded by the y-axis and the graphs of y=x3/1+x2 and y=4-2x.
The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of the solid.
Volume= ∫1.48770 (4-2x- x3/(1+x2) )2 dx = 8.997
Substitution
∫ tan x dx (Hint: Let u=cos x )
ln |sec x| +C
Find f(x) by solving the separable differential equation
dy/dx = 3x2+1/2y with the initial condition f(1)=4
f(x)= (x3+x+14)1/2
A particle moves along the x-axis with velocity at time t on the interval
[0,+) given by v(t)= -1+e1-t
Find all values of t at which the particle changes direction.
v(t)=0 when 1=e1-t, so t=1.
v(t) > 0 for t<1 and v(t)<0 for t>1.
Therefore, the particle changes direction at t=1.