First Midterm
Find all vertical Asymptotes:
F(x)= [5(x^2-7x+10)]/[x(x^2-25)]
x=-5,0
Find the Derivative of f(x)=
(5/x^3) - (2/x^2) - (1/x)+200
(-15/x^4) + (4/x^3) + (1/x^2)
Find the linearization L(x) of f(x)= x^3+1 at x=2.
L(x)= 12x -15
Initial value problem: find f(x) given f'(x)=x+1 and f(2)=6.
(1/2)x^2+x+2
Evaluate the limit:
lim(x->∞) (e^x)/(x^2)
∞
Evaluate h"(x) if h(x)= x^2*e^4
2e^4
Let y=-2x^2+3. find the differential dy if x changes from 2 to 1.5
4
Find the indefinite integral: ∫ (2sin√x)÷√x
-4cos√x +c
An object is moving with a(t)=4 m/s^2, v(0)=-6m/s and s(0)=5m.
Find v(t) and s(t).
v(t)= 4t-6
s(t)= 2t^2-6t+5
Substitution Rule:
∫ √(lnx)/x dx
2/3 ln^(3/2) +c
Find the limit:
lim(x->0) (1+ 1/x)^x
∞^0 and limit=1.
A ladder of 8m is leaning on a wall, the foot of the ladder is moving at 2m/s away from the wall, and the foot of the ladder is 4m away from the wall. Find the rate at which the ladder is going down the wall.
-2/√3 m/s
Use average value function f(x)=cos x on [-π/2, π/2]
2/π
Evaluate lim (x->∞) (e^5x+x)^(1/x)
e^6
Use substitution Rule to evaluate the definite integral:
∫ (cos 2x)^3 * sin 2x dx on [0,π/4].
1/8
Find the derivative f'(x) of f(x)= x^π + π^x
πx^(π-1) + π^xlnπ
Find the number c that satisfies the mean value theorem for f(x)=x^3+x-1 on [0,2]
remember: f'(c)= (f(b)-f(a)) / (b-a)
√(2/3)
Write a definite integral that represents the region bound the following curves:
y=x and y=x^2-2
∫ [x-(x^2-20] dx on (-1,2)
A rectangular garden with an area 80m^2 is surrounded by a grass border of 1m on two sides and 2m on the other. Find the minimum dimensions of the garden and the border.
x= 2√10 and y=4√10
Use Riemann sums and the definition of definite integral to find the area of f(x)=4-x^2 over [0,2].
16/3
lim (x->∞) (1+3/x)^3x
e^9
Air is pumped into a spherical balloon so that volume increases at 100 cm^3/s. The diameter is 50 cm, how fast is the balloon's radius increasing?
V= 4/π^3
1/25π cm/s
Find indefinite integral:
∫ 6∛x+ (x+1)/x dx
9/2x^(4/3) +x+ ln|x| +c
Use the definition of definite integrals and right Riemann Sums to evaluate ∫ (x^3-1) on [0,2].
2
A square-based box (cuboid) with volume 16m^3, uses cost "p" dollars for the side materials, and "2p" for the square base. What are the minimum dimensions of the crate?
length=width=2 ft bc square base and height= 4 ft.