8/9/2019 Derivatives _Word Problems

http://slidepdf.com/reader/full/derivatives-word-problems 1/5

Q.1 The gradient function of a curve y =ƒ(x) is given by ƒ′

(x) =4x −5. The curve passes through the point (2, 3). Find

the equation of the curve.

Q.2 The graph has a horizontal point of inflexion at A, a

point of inflexion at B and a maximum turning point at C.

Sketch the graph of f ‘(x)

Q.3 The diagram shows a window consisting of two sections. The top section is a semicircle

of diameter x m. The bottom section is a rectangle of width ‘x’ metres and height ‘y’ metres.

The entire frame of the window, including the piece that separates the two

sections, is made using 10 m of thin metal.

The semicircular section is made of coloured glass and the rectangular

section is made of clear glass. Under test conditions the amount of light

coming throughone square metre of the coloured glass is 1 unit and the

amount of light coming throughone square metre of the clear glass is 3

units. The total amount of light coming through the window under test

conditions is L units.

Q.3 The diagram on the right hand side shows points A, B, C and

D on the graph y = f(x). At what points is f ’(x) > 0 and f ”(x) = 0

a A

b B

c C

d D

8/9/2019 Derivatives _Word Problems

http://slidepdf.com/reader/full/derivatives-word-problems 2/5

Q.4 The diagram shows the function f(x). Which of the following is true ?

a f ‘ (a) > 0 and f “ (a) < 0

b f ‘ (a) > 0 and f “ (a) > 0c f ‘ (a) < 0 and f “ (a) < 0

d f ‘ (a) < 0 and f “ (a) > 0

Q.5 Find the derivative of x2.ex 

Q.6 The gradient of a curve is given by 6x-2.The curve passes through the point (-1,4).Find

the equation of the curve.

Q.7 Let f(x) = x2 – 3x + 2

Find the coordinates of the stationary points of f(x) and determine their nature.

Hence sketch the graph of y=f(x) showing all stationary points and y-intercept

Q.8 The graph y=f(x) in the diagram has a

stationary point when x = 1 , a point of

inflexion when x=3 and a horizontal asymptote

y = -2

Sketch the graph y = f ‘(x) clearly indicating its

feature at x=1 and at x =3

Q.9 Differentiate x2 .tanx

Q.10 Let f(x) = 1 + ex

Show that f(x)* f(-x) = f(x) + f(-x)

Q.11 A rainwater tank is to be designed in the shape of a cylinderwith radius r metres and height h metres. The volume of the tank is

to be 10 cubic metres. Let A be the surface area of the tank,

including its top and base, in square metres.

Given that A = 2 rπ2  + 2 rh , Show that A = 2 rπ π

2  +

20

r 

8/9/2019 Derivatives _Word Problems

http://slidepdf.com/reader/full/derivatives-word-problems 3/5

Show that A has a minimum value and find the value of ‘r’ for which the minimum value

occurs.

Use volume of a cylinder = rπ2h

Q.12 Let ƒ (x) = (x + 2).(x2 + 4)

Show that the graph of f(x) has no stationary points.

Find the value of ‘x’ for which the graph y = f(x) is concave down and for what values it is

concave up.

Sketch the graph y=f(x) indicating y and x intercepts

Q.13

Differentiate with respect to ‘x’

a) x.sinx

b) (ex + 1)2

Q.14

8/9/2019 Derivatives _Word Problems

http://slidepdf.com/reader/full/derivatives-word-problems 4/5

8/9/2019 Derivatives _Word Problems

http://slidepdf.com/reader/full/derivatives-word-problems 5/5

Q.20