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(page 1.4 )
Expand by Pascal’s triangle.
According to row 10 of Pascal’s triangle, row 11 can be obtained as follows:
(page 1.7 )
Expand each of the following. (a)
(b)
(a)
(b)
(page 1.7 )
(a) Expand.
(b) Using the result of (a) , expand.
(a)
(b)
(By the result of (a) )
(page 1.8 ) (a) Expand.
(b) Using the result of (a) , expand.
(a) (b)
(By the result of (a) )
(page 1.9 )
Expand each of the following.
(a)
(b) (c)
(a)
(b)
(c)
(page 1.10 ) Find the coefficients of the terms or the constant terms as specified in the expansions of the following. (a) Term in of (b) 3rd term of in ascending powers of x (c) Constant term of
(a) General term of the expansion Consider the term in , i.e. take. F 0 5 CCoefficient of
(b) General term of the expansion 3rd term F 0 5 CCoefficient of the 3rd term
(c) General term of the expansion
When (i.e. ), the term is a constant term. F 0 5 CConstant term
(page 1.11 ) It is given that F 0 2 Bterms involving higher powers of x , where q is a non-zero real number and n is a positive integer. Find the values of n and q.
By comparing the coefficients of x and ,
From (2),
or (rejected)
Substitute into (1),
(rejected) or
(page 2.8 )
Evaluate each of the following limits.
(a)
(b) (c)
(d)
(a)
(b)
(c)
(d)
(page 2.9 )
Evaluate each of the following limits.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(page 2.13 ) It is given that the function. If there is an increment at , find the value of F 0 4 4 y.
(page 2.14 ) For , evaluate F 0 4 4 y and for each of the following increments F 0 4 4 x at. (a) (b)
(a)
(b)
(page 2.14 ) Find F 0 4 4 y and for each of the following functions. (a) (b)
(a) Let.
(b) Let.
(page 3.4 ) Differentiate each of the following functions with respect to x. (a) (b)
(a) (b)
(page 3.4 ) Differentiate each of the following functions with respect to x.
Find the derivative of each of the following functions with respect to x. (a) (b)
(a)
(b)
(page 3.15 )
Differentiate each of the following functions with respect to x. (a) (b)
(a)
(b)
(page 3.16 )
Find if
(a).
(b).
(a)
By the chain rule,
(b)
(page 3.20 )
Find of the following functions.
(a)
(b)
(a)
(b)
(page 3.20 )
Find of the following functions.
(a)
(b)
(a) (b)
(page 3.20 ) Find of the following functions. (a) (b)
(a) Differentiating x with respect to y ,
(b) Differentiating x with respect to y ,
(page 3.23 ) Find and of the following functions. (a) (b)
(a)
(b)
(page 3.23 ) Find of the following functions. (a) (b)
(a)
(b)
(page 3.24 ) If , prove that .
(page 4.4 )
Find the equation of the tangent to the curve at the point (4, 7).
Slope of the tangent at (4, 7)
The equation of the tangent is
(page 4.4 )
Find the equations of the tangents to the curve which are parallel to the line.
Let be the point of contact of a required tangent.
Slope of the given line
Differentiate both sides of the equation of the given curve,
Slope of the tangent at
or or
When ,
F 0 5 Cis a point of contact.
The equation of the tangent at is
When ,
F 0 5 Cis a point of contact. The equation of the tangent at is
(page 4.5 ) Find the equations of the tangents to the curve which pass through the point (3, F 0 2 D15).
Let be the point of contact of a required tangent. Since lies on the curve,
Slope of the tangent Slope of the line joining and (3, F 0 2 D15)
(1) F 0 2 D(2):
F 0 5 C or When ,
F 0 5 C(1, 3) is a point of contact. Slope of the tangent F 0 5 CThe equation of the tangent at (1, 3) is
When ,
F 0 5 C(5, F 0 2 D65) is a point of contact.
Slope of the tangent F 0 5 CThe equation of the tangent at (5, F 0 2 D65) is
(page 4.9 ) A particle moves along the y -axis so that its displacement at any time t (in seconds) is given by where. Find the velocity and acceleration, in magnitude and direction, at.
pill for 3 hours.
(page 4.11 )
The rate of increase of the volume of a hemispherical bubble is. What are the rates of change of its radius and its curved surface area when the radius is 2 cm?
Suppose at time t seconds, the radius of the bubble is r cm. Its volume is given by , and its curved surface area A is given by.
Then and.
Given that , when ,
F 0 5 CWhen the radius is 2 cm, the rate of change of the
radius is and the rate of change of the
curved surface area is.
(page 4.19 )
Find the local maximum and minimum of.
F 0 5 CWhen , or. x x F 0 3 CF 0 2 D 2 x F 0 3 DF 0 2 D 2 F 0 2 D 2 F 0 3 C x F 0 3 C 5 x F 0 3 D 5 x F 0 3 E 5 f ( x ) (^68) F 0 2 D 275 f '( x ) (^) F 0 2 B 0 F 0 2 D 0 F 0 2 B
F 0 5 CThe local maximum is 68 and the local minimum
is F 0 2 D275.
(page 4.20 ) Find the turning points of the curve.
F 0 5 CWhen , or. x y F 0 2 D 0 F 0 2 B 0 F 0 2 D
F 0 5 Cis a maximum point and
is a minimum point.
(page 4.21 ) Find the turning point of the curve.
F 0 5 CWhen ,.
x y 6 F 0 2 B 0 F 0 2 D
F 0 5 Cis a maximum point.
(page 4.26 ) Find the global extrema of the function for.
When ,
or (rejected) x (^) x F 0 3 D 0 0 F 0 3 C x F 0 3 C 3
x F 0 3 D 3 3 F 0 3 C x F 0 3 C 4 x F 0 3 D 4
f ( x ) (^100) F 0 2 D 35 F 0 2 D 4 f '( x ) F 0 2 D 0 F 0 2 B
F 0 5 CThe global maximum of f ( x ) is 100, the global
minimum of f ( x ) is F 0 2 D35.
(page 4.27 )
Find the global extremum of the function for.
When ,
or (rejected) x (^) F 0 2 D 10 F 0 3 C x F 0 3 C F 0 2 D 6
x F 0 3 DF 0 2 D 6 F 0 2 D 6 F 0 3 C x F 0 3 CF 0 2 D 4
f ( x ) 96 f '( x ) (^) F 0 2 B 0 F 0 2 D
F 0 5 CThe global maximum of f ( x ) is 96.
(page 4.28 )
The cost $ C of delivering a batch of goods by x trucks is , where. How many trucks should be used to minimize the cost? What is the lowest cost?
When ,
or x F 0 3 DF 0 2 D 7 (rejected) x x F 0 3 D 5 5 F 0 3 C x F 0 3 C 7 x F 0 3 D 7 7 F 0 3 C x F 0 3 C 1 0
x F 0 3 D 10
C 5 920 5 600 5 960
F 0 2 D 0 F 0 2 B
When , C attains its least value. F 0 5 C7 trucks should be used to minimize the cost.
The lowest cost is $5 600.
(page 4.29 ) A farmer is planning to build a warehouse with the area of. If the width of the warehouse is x m, then the cost of building the warehouse is C ( x ) (in thousand dollars). Given that , where , find the lowest cost of building the warehouse. (Give your answer correct to the nearest thousand dollars.)
When ,
or (rejected) x C '( x ) (^) F 0 2 D 0 F 0 2 B
F 0 5 CWhen , C ( x ) attains its least value.
(corr. to the nearest integer) F 0 5 CThe lowest cost of building the warehouse is
566 thousand dollars.
(page 4.30 ) There is a wire of 40 cm long. If the wire is cut into two parts and each part is bent into a square, find the least value of the sum of the areas of the two squares.
(page 5.13 )
Find the following limits.
(a)
(b)
(c) (d)
(Give your answers correct to 3 significant figures if necessary.)
(a)
(corr. to 3 sig. fig.)
(b)
(corr. to 3 sig. fig.)
(c)
(d)
(page 5.16 )
A principal of $80 000 is deposited in a bank at an interest rate of 0.5% per quarter, compounded continuously.
(a) Find the amount after 2 years.
(b) Find the interest obtained after 10 years.
(Give your answers correct to the nearest dollar.)
(a) Amount
(corr. to the nearest dollar)
(b) Amount
(corr. to the nearest dollar) F 0 5 CInterest
(page 5.17 )
For a principal of $70 000 deposited in a bank at an interest rate compounded continuously, an interest of $5 000 will be obtained after 4 years. Find the interest rate p.a. (Give your answer correct to 3 significant figures.)
Let r % be the required interest rate p.a.
(corr. to 3 sig. fig.)
F 0 5 CThe required interest rate p.a. is 1.72%.
(page 5.17 ) The number of bees in a place increases at a constant rate continuously. It increases by 80% after 2 years. Find the growth rate per month. (Give your answer correct to 3 significant figures.)
Let be the original number of bees, and r % be the growth rate per month. Then (corr. to 3 sig. fig.) F 0 5 CThe growth rate per month is 2.45%.
(page 5.18 ) Referring to Example 5.9, the mass of a substance will be 200 grams after 30 years and it will be 40 grams after 100 years. (a) Find the value of k. (Give your answer correct to 4 decimal places.) (b) What is the initial mass of the substance? (Give your answer correct to the nearest gram.) (c) Find the mass of the substance after 200 years. (Give your answer correct to the nearest gram.) (d) How long will it take for the mass to be 5% of the initial mass? (Give your answer correct to 1 decimal place.)
(a) For , when.
when.
(corr. to 4 d.p.)
(b) Substitute into (1),
(corr. to the nearest integer)
F 0 5 CThe initial mass is 399 grams.
(c) When ,
(corr. to the nearest integer) F 0 5 CThe mass of the substance will be 4 grams
after 200 years.
(d) When ,
(corr. to 1 d.p.) F 0 5 CIt will take 130.3 years for the mass to be 5%
of the initial mass.
(page 5.22 )
It is known that the set of data below follows a linear relation connecting x and y where a and b are constants.
x 1 2 3 4 5 y (^8 5 2) F 0 2 D 1 F 0 2 D 4
(a) Find the equation connecting x and y.
(b) Find y if.
(a) From the given set of data, we have a linear graph.
Choosing (1, 8) and (5, F 0 2 D4), we have Slope When ,.
F 0 5 CThe required equation is.
(b) From the equation obtained in (a) , when ,
(page 5.23 ) It is known that the set of data below follows the relation connecting x and y where A and n are constants. x 1 3 5 7 9 y 6 162 750 2 058 4 374 (a) Find the equation connecting x and y. (b) Find y if.
(a) By taking common logarithm on both sides of , we have
Hence we convert the given data into the following. log x 0 0.477 0.699 0.845 0. log y 0.778 2.210 2.875 3.313 3.
Let and , then (*) becomes.
the following functions.
(a)
(b)
(a)
(b)
(page 6.8 )
Find the derivative with respect to x of each of the following functions. (a)
(b)
(a)
(b)
(page 6.8 )
Find the derivative with respect to x of each of the following functions.
(a)
(b)
(a)
(b)
(page 6.9 )
Find the derivative with respect to x of each of the following functions.
(a)
(b)
(a)
(b)
(page 6.10 )
Find the derivative with respect to x of each of the following functions. (a)
(b)
(a)
(b)
(page 6.12 ) If , find.
Taking the natural logarithm on both sides of the equation, we have
Differentiate both sides with respect to x , we have
(page 6.12 ) If , find.
Taking the natural logarithm on both sides of the equation, we have
Differentiate both sides with respect to x , we have
(page 6.13 ) If , find.
(page 6.14 ) If , find.
(page 6.16 ) If , find (a). (b) the equation of the tangent to the curve
at.
(a)
(b) When ,
F 0 5 CThe equation of the tangent is
(page 6.17 )
The profit P ( x ) (in thousand dollars) from selling x units of a product can be modelled by. Find the rate of change of the profit with respect to the number of units sold when 15 units are sold. (Give your answer correct to 3 significant figures.)
(corr. to 3 sig. fig.) F 0 5 CThe required rate of change is
0.876 thousand dollars / unit.
(page 6.17 )
The temperature B ( t ) (in F 0 B 0C) of a bottle of water after cooling for t minutes is modelled by. Find the rate of change of water temperature with respect to time after cooling for 6 minutes. (Give your answer correct to 3 significant figures.)
(corr. to 3 sig. fig.)
F 0 5 CThe required rate of change of water temperature
with respect to time is F 0 2 D3.63 F 0 B 0C / minute.
(page 6.18 ) If , find the turning point(s) of the graph of.
When ,
F 0 5 CThe graph of has a minimum point at.
F 0 5 CThe minimum point is.
(page 6.18 )
If , find the local extremum of f ( x ).
When ,
or
F 0 5 C f ( x ) attains its local minima at and.
F 0 5 CThe local minimum of f ( x ) is.
(page 6.19 ) The number of fish P ( t ) in a fish pond after t months can be modelled by. Find the maximum number of fish in the fish pond. (Give your answer correct to the nearest
(page 7.14 )
Find.
Let.
Then.
When ,.
(page 7.15 )
Find.
Let.
Then.
(page 7.16 )
Find.
Let. Then.
(page 7.16 )
Find.
Let.
Then.
(page 7.17 )
(a) If , find the values of constants A and B. (b) Hence find.
(a)
i.e.
(b)
(page 7.20 ) After t months of installation of a vending machine, the rate of change of the total profit P ( t ) (in thousand dollars) made with respect to time can be modelled by. (a) Find the function of the total profit. (b) Find the total profit made by the vending machine in the first half year. (Give your answer correct to 3 significant figures.)
(a)
(b) (corr. to 3 sig. fig.) F 0 5 CThe total profit made by the vending machine
in the first half year is 33.9 thousand dollars.
(page 7.21 ) A manager of a shop notices that the weekly sales is declining, so he starts a promotion plan to boost the weekly sales. After t weeks since the start of the plan, the rate of change (in thousand dollars/week) of the weekly sales of the shop can be modelled by. It is given that the weekly sales is 20 thousand dollars at the start of the plan (i.e. ). (a) Find the function of the weekly sales after t weeks since the start of the plan. [ Hint: Let. ]
(b) Find the weekly sales after 10 weeks since the start of the plan. (Give your answer correct to 3 significant figures.)
(a) Let be the function of the weekly sales after t weeks since the start of the plan, where S ( t ) is in thousand dollars.
Let. Then.
(b) (corr. to 3 sig. fig.) F 0 5 CThe weekly sales is 23.5 thousand dollars
after 10 weeks since the start of the plan.
(page 7.22 )
The slope at any point ( x , y ) of a curve is. If the y -intercept of the curve is F 0 2 D2, find the equation of the curve.
F 0 5 1The y -intercept of the curve is F 0 2 D2. F 0 5 CThe curve passes through (0, F 0 2 D2).
Substitute (0, F 0 2 D2) into ,
F 0 5 CThe equation of the curve is.
(page 7.22 ) At any point on a certain curve,. If the slope of the curve at (3, 0) is 7, find the equation of the curve.
At the point (3, 0),
i.e.
F 0 5 1The curve passes through (3, 0).
F 0 5 CThe equation of the curve is.
(page 8.8 ) Evaluate.
(page 8.8 ) Evaluate.
(page 8.8 ) Evaluate.
(page 8.9 ) Evaluate.
(page 8.9 )
(a) Q For , F 0 5 CRequired area
(b) Q For , F 0 5 CRequired area
(c) Required area
(page 9.5 )
Find the area of the region between the curve and the x -axis from to.
Required area
(page 9.6 )
Find the area of the region bounded by the curve and the lines and.
Rewrite the equation as , i.e.. As shown in the figure, the curve is split into two parts, and. The required area is the sum of and. Since the curve is symmetrical about the x -axis,.
F 0 5 CRequired area
(page 9.7 )
Find the area of the region bounded by the curve for , the y -axis and the lines and.
Rewrite the equation as. Required area
(page 9.9 ) Find the area of the region between and from to.
Required area
(page 9.10 ) Find the area of the region bounded by the curve and the line from to.
Required area
(page 9.10 )
Find the area of the region bounded by the curves and.
By solving the simultaneous equations and , the points of intersection ( F 0 2 D1, F 0 2 D1) and ( F 0 2 D1, 1) can be obtained. Required area
(page 9.11 )
Find the area of the region bounded by the curve and the line.
By solving the simultaneous equations and , the points of intersection ( F 0 2 D2, 1) and (3, F 0 2 D4) can be obtained. Rewrite the equations as and.
Required area
(page 9.11 )
In the figure, and intersect at A , O and B. Find the total area of the shaded regions.
By solving the simultaneous equations and , the points of intersection ( F 0 2 D3, 21), (0, 0) and (4, 0) can be obtained. Required area
(page 9.19 ) Use the trapezoidal rule with 4 subintervals to estimate. (Give your answer correct to 4 decimal places.)
Let.
(corr. to 4 d.p.)
(page 9.20 ) It is given that , where and. The figure shows a sketch of against x.
(a) Estimate the value of g (3) by using the trapezoidal rule with 4 subintervals. (Give your answer correct to 4 decimal places.) (b) From the figure, determine whether the approximate value obtained in (a) is an underestimate or an overestimate.
(a) x 1 1.5 2 2.5 3 g '( x ) 1.271 8 3
1.672 2 5
2.477 8 1
4.045 6 2
7.025 66
(corr. to 4 d.p.)
(corr. to 4 d.p.)
(b) Since the graph of g '( x ) is concave upwards on the interval , the approximate
