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Evaluate the following limits
(a) Lim y"\\to" -1 Fourth root of 4y^3 minus 3
(b) lim x"\\to" infinity 1+x all over 1-x
(c) lim x "\\to" 2 x - 2 all over 4 - x^2
(d) lim x "\\to" 0 sinx - x cos x all over x^3
Evaluate the following limits
(a) Lim y"\\to" -1 Fourth root of 4y^3 minus 3
(b) lim x"\\to" infinity 1+x all over 1-x
(c) lim x "\\to" 2 x - 2 all over 4 - x^2
(d) lim x "\\to" 0 sinx - x cos x all over x^3
a) find and classify the critical points of the functions f(x) = 2x^3 + 3x^2 - 12 x +1 into maximum, minimum and inflection points as appreciate.
(b) The sum of two positive numbers is S. find the maximum value of their product.
a) find and classify the critical points of the functions f(x) = 2x^3 + 3x^2 - 12 x +1 into maximum, minimum and inflection points as appreciate.
(b) The sum of two positive numbers is S. find the maximum value of their product.
Determine whether ~u and ~v are orthogonal vectors, make an acute or obtuse angle: (1.1) ~u =< 1, 3, −2 >, ~v =< −5, 3, 2 >. (2) (1.2) ~u =< 1, −2, 4 >, ~v =< 5, 3, 7 >.
how many terms of the arithmetic series 10 + 8 + 6 +... will make the sum 24?
Performance Task: Beyond Walls
The Congressman of your town will visit your school for the inauguration of the newly constructed building. You are a painter who is famous for designing regions bounded by curves, that is why you are asked to present your geometric pattern design and computations for the cost of paint which will be used by the Principal. Your output will be subject for approval based on the criteria:
1. illustration of design (5 points)
2. accuracy (5 points)
3. exactness of solutions (5 points)
4. overall presentation (5 points)
Perform the following:
1. State your proposed function and graph
2. Solve the area of the regions bounded by the two functions
3. Use this ratio to estimate the paint needed and to compute the cost of paint 1 square unit = 10 sq. ft. 1 liter of paint (cost P200) can cover 12 sq. ft.
4.The supply S and demand D for a particular commodity satisfy the equations
S(p)=100+p+pt and D(p)=200-p-pt
- Find the equilibrium price p(t) at any time t
- Find the long range equilibrium price
- graph the particular solution p(0)=75 and p(0)=25 hence discuss the long term behaviour of the price of this commodity
5.find a series solution in powers of x of the equation
2x2d2y/dx2+xdy/dx+(x2-1)y=0
1.The yield y(t) (in bushes)per acre of a corn crop satisfies the equation dy/dt+y=100+e-t
if y(0)=0 find y at any time t
2.The population N(t) of a species of micro-organism in a laboratory setting at any time t is established to vary under the influence of a certain chemical at a rate given by dN/dt=t-2et show that N(t)=t-2et+c.hence if N(0)=400 determine the population when t=5
3.A single individual starts a rumour in a community of 200 people. the rumour spread at a rate proportional to the number of people who have not yet heard the rumour. After 2days 10 people have heard the rumour
- how many people will have heard the rumour after 5days
- how long will it take for the rumour to spread to 100 people
The yield y(t) (in bushes)per acre of a corn crop satisfies the equation dy/dt+y=100+e-t
if y(0)=0 find y at any time t
