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6.02 Applications of the differentiation of exponential functions

Worksheet
Tangents and normals
1

For each of the following curves and given point:

i

Find the equation of the tangent.

ii
Find the equation of the normal.
a

f \left( x \right) = 2 e^{x} at the point where x = 0.

b

f \left( x \right) = 5e^{4x} at the point (0, 5).

c

f \left( x \right) = \dfrac{e^{x}}{6} at the point \left(0, \dfrac{1}{6}\right).

d

f \left( x \right) = 4 e^{3x} - 2x at the point where x=0.

2

For each of the following curves, at the given point:

i

Find the exact gradient of the normal.

ii
Find the equation of the normal.
a

y=\dfrac{e^x}{x} at the point where x=-1.

b

y=xe^x at the point where x=1.

c

y=x-e^{-x} at the point where x=2.

3

The tangent to the curve f \left( x \right) = 2.5 e^{x} at the point x = 7.5, is parallel to the tangent to the curve g \left( x \right) = e^{ 2.5 x} at the point where x = k.

Find the value of k.

4

Consider y = x e^{ - x }:

a

Find the gradient function.

b

Find the gradient of the normal at the point \left(3, 3 e^{ - 3 }\right).

c

Hence, find the equation of the normal at the point \left(3, 3 e^{ - 3 }\right).

5

The tangent to the curve y=x^{2}e^{x} at x=1, cuts the axes at points A and B .

a

Find the exact gradient of the tangent at x=1.

b
Find the equation of the tangent at x=1.
c
Find the coordinates of point A, where the tangent cuts the x-axis
d
Find the coordinates of point B, where the tangent cuts the y-axis
e
Determine the exact area of triangle AOB where point O is the origin.
6

Consider the functions y = 3 e^{-x} and y = 2 + e^{-x}:

a

Find the point of intersection of the functions.

b

Find the gradient of the tangent to y = 3 e^{-x} at the point of intersection.

c

Determine the acute angle this tangent makes with the x-axis, to the nearest degree.

d

Find the gradient of the tangent to y = 2 + e^{-x} at the point of intersection.

e

Determine the acute angle this tangent makes with the x-axis, to the nearest degree.

f

Find the acute angle between the two tangents.

7

Consider the functions g \left( x \right) = e^{ 7 x} and f \left( x \right) = e^{ - 7 x }.

a

Consider the tangent line to g \left( x \right) at x = 0. Find the acute angle that the tangent line makes with the x-axis, correct to one decimal place.

b

Consider the tangent line to f \left( x \right) at x = 0. Find the acute angle that the tangent line makes with the x-axis, correct to one decimal place.

c

Hence, determine the acute angle between the two tangents at x = 0.

8

Consider the curve f \left( x \right) = e^{x} + e x. Show that the tangent to the curve at the point \left(1, 2 e\right) passes through the origin.

Exponential graphs
9

Consider the function f \left( x \right) = 3 - e^{ - x }.

a

Find f \left( 0 \right).

b

Find f' \left( 0 \right).

c

State whether f (x) is an increasing or decreasing function. Explain your answer.

d

Find the limit of f (x) as x \to \infty.

10

Consider the function f \left( t \right) = \dfrac{4}{2 + 3 e^{ - t }}.

a

Find f \left( t \right) when t = 0.

b

Find f' \left( t \right).

c

State whether f(x) is an increasing or decreasing function. Explain your answer.

d

Hence, state how many stationary points the function has.

e

Find the limit of f (t) as t \to \infty.

11

Consider the function y = e^{x} \left(x - 3\right).

a

Find the coordinates of the turning point.

b

State whether this is a minimum or maximum turning point.

12

Consider the function f \left( x \right) = 4 e^{ - x^{2} }.

a

Find f' \left( x \right).

b

Find the values of x for which:

i
f' \left( x \right) = 0
ii

f' \left( x \right) \gt 0.

iii

f' \left( x \right) \lt 0

c

Find the limit of f(x) as x \to \infty.

d

Find the limit of f(x) as x \to - \infty.

e

Sketch the graph of f \left( x \right).

13

Consider the function f \left( x \right) = e^{x} - 2 e^{ - x }.

a

Find the x-intercept, to two decimal places.

b

Find the y-intercept.

c

What value does y approach as x \to \infty.

d

What value does y approach as x \to - \infty.

e

Sketch the graph of the function.

14

For each of the following functions:

i

Find the x-intercept.

ii

Find the y-intercept.

iii
Find the coordinates of any stationary points.
iv

State the nature of the stationary points.

v

Find any points of inflection.

vi

Describe what happens to the function as x \to \infty.

vii

Sketch the graph of the function.

a
y = x e^{-x}
b

y = e^{2x} - 3 e^{x}

15

Consider the function y = x^{3} e^{-x}:

a

Find the x-intercept.

b

Find the y-intercept.

c

Find y \rq.

d

Find the coordinates of any stationary points, correct to two decimal places if necessary.

e

Describe the nature of the stationary points.

f

Sketch the graph of the function.

16

For the function y = e^{-x}\sqrt{x}:

a

Find the x-intercept.

b

Find the y-intercept.

c

Find y \rq.

d

Find the coordinates of any stationary points, correct to two decimal places.

e

State the nature of the stationary points.

f

Sketch the graph of the function.

17

Consider the function y = e^{ 2 x} - 3 e^{x}.

a

Find the x-intercept.

b

Find the y-intercept.

c

Find the coordinates of any stationary points, correct to two decimal places.

d

State the nature of the stationary point.

e

Describe the behaviour of this function as x \to \infty and x \to -\infty.

f

Sketch the graph of the function.

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Outcomes

MA12-3

applies calculus techniques to model and solve problems

MA12-6

applies appropriate differentiation methods to solve problems

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