Revision simple differentiation exercises 1. 2. Differentiate
with respect to x Differentiate with respect to x: (a) (a) 3 4x 2 2
(x + l) (b) (b) e sin x
1n(3x 1).
3.
(a)
Find the equation of the tangent line to the curve y = ln x at
the point (e, 1), and verify that the origin is on this line. d dx
(x ln x x) = ln x. Show that
(b)
(c)* The diagram shows the region enclosed by the curve y = ln
x, the tangent line in part (a), and the line y = 0. y
1
( e ,
1
)
0
1
2
3
x
Use the result of part (b) to show that the area of this region
is 4. 3 Let f (x) = x . f (5 + h) f (5) h Evaluate for h = 0.1.
1 2
e 1.
(a)
(b) 5.
f (5 + h) f (5) h What number does approach as h approaches
zero? y=e 2x cos x.
The diagram shows part of the graph of the curve with
equation
1
y
P a , (
b
)
0
x
(a) ***
dy 2x Show that dx = e (2 cos x sin x).
(b)
d2 y 2 Find dx .
There is an inflexion point at P(a, b). 3 tan a = 4 ;
(c)
Use the results from parts (a) and (b) to prove that: 2x the
gradient of the curve at P is e .
(i)
6. 7.
2 Given the function f(x) = x 3bx + (c + 2), determine the
values of b and c such that f(i) = 0 and f (3) = 0. Consider the
function f(x) = k sin x + 3x, where k is a constant. (a) Find f
(x).
(b) 8.
When x = 3 , the gradient of the curve of f(x) is 8. Find the
value of k.x
3 2 Let f(x) = e + 5cos x. Find f (x).
9.
3 2 Let f(x) = 6 x . Find f (x).
10.
The population p of bacteria at time t is given by p = 100e (a)
(b) the value of p when t = 0; the rate of increase of the
population when t = 10.
0.05t
. Calculate
2
11.
Part of the graph of the periodic function f is shown below. The
domain of f is 0 x 15 and the period is 3.f( x ) 4 3 2 1 0 x
0
1
2
3
4
5
6
7
8
9
1 0
(a) (b) 12.
Find:
(i)
f(2);
(ii)
f (6.5);
(iii)
f (14).
How many solutions are there to the equation f(x) = 1 over the
given domain?
Let f(x) = 1 + 3 cos(2x) for 0 x , and x is in radians. (a) (i)
(ii) Find f (x). Find the values for x for which f (x) = 0, giving
your answers in terms of .
The function g(x) is defined as g(x) = f(2x) 1, 0 x 2 . (b) (i)
(ii) 13. The graph of f may be transformed to the graph of g by a
stretch 1in the x-direction with scale factor followed by another
transformation. Describe fully this other transformation. Find the
solution to the equation g(x) = f(x) (a) Write down: (i) f (x);
(ii) f (0).
2 The function f is defined by f : x 0.5 x + 2 x + 2.5.
Let N be the normal to the curve at the point where the graph
intercepts the y-axis. Show that the equation of N may be written
as y = 0.5x + 2.5. g : x 0.5 x + 2.5 Let (c) (i) (ii) Find the
solutions of f(x) = g(x). Hence find the coordinates of the other
point of intersection of the normal and the curve.
(b)
(d)* Let R be the region enclosed between the curve and N. (i)
(ii) Write down an expression for the area of R. Hence write down
the area of R.
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