## Derivative Exercises for Intermediate Level

### 1. Equation of the tangent line :        Solution 1

Given the quadratic equation y = 3x2 + 2 and the math function
y = (1/2)x3 - 1, you are asked to :

a) figure out the derivative y' of both functions using the
derivative rules,

b) figure out, for both curves, the slope of their tangent line at x = 1
using derivatives,

c) find the equation of both tangent lines,

d) to show both curves and their tangent line on the same graph
in order to visualize those equations and by the way check that your
answers make sense. Use the free online Graphing Calculator to plot
the graph of these functions online. You will find there a free online graph plotter
and an online scientific calculator as well.

### 2. Derivative, tangent line and trigonometric function :        Solution 2

You are asked to calculate the derivative of the following mathematical function : Next, we ask you to use the derivatives to find the cartesian equation of
the tangent line to the curve f(x), at x = 0.

### 3. Composition of functions with logarithm :        Solution 3

Derivate the composite function using the derivative rules.

How can you check whether your answer is correct or not ?

Use the free graph plotter online which is a real online graphing calculator
(with an online scientific calculator for free too) that enables you to trace
your mathematical functions on an XY coordinate grid.
How to procede ? Enter the equation of y in First Graph and check the radio button
Derivative of the Graph Plotter. This way, the derivative of the function y
will be computed by the program and will be graphed by clicking on
the button Draw.
Next, in Second Graph, enter the equation of the derivative y ' that you have
figured out. This way you will trace on the same graph your answer and the program
answer. If it is imposible to distinguish one curve from the other, this means that

### 4. Tangent line parallel to the bisector of the first quadrant :    Solution 4

Given the equation of a curve f(x) = 2x3 - 6x + 5, you are asked to
find the coordinates of the points where the tangent lines are parallel
to the bisector of the first quadrant.

You are also asked to figure out the equation of those tangent lines and
to plot the graph of the curve and its tangent lines.

Use our graphing tool to plot a graph online : Free Online Graphing Calculator.
The objective of this is that you visualize your equations and by the way that

### 5. The derivative of the position x(t), with respect to time t, is the instantaneous velocity (or instantaneous speed) v(t):      Solution 5

A particle is moving along a straight line X according to the following
equation : x(t) = 2t2 + 8t + 9, where x(t) represents (in meters)
the distance of the particle at time t, from the point of departure.
You are asked to find the position of the particle and its velocity :

a) at time t = 0 sec,

b) at time t = 1 sec.

### 6. Derivative, position and velocity :        Solution 6

A particle moves along a straight line. Its position, x, as a function of time
is given by the equation x(t) = 2t3 - 12t2 + 20t + 3 where x(t) represents
the distance (in meters) traveled at time t. You are asked to :

a) calculate the position of the particle at time t = 2 sec and to figure out
its velocity at that instant,

b) calculate the position of the particle if its velocity has reached v = 2 m/sec

### 7. Derivative, coordinates of a point in a cartesian coordinate system, and tangent line :         Solution 7

Find the coordinates of the points on the curve y = x3 - 12x + 1 where the tangent
line to the curve has a slope equal to zero.

After you have figured out the answer, use the free online Graph Plotter
to trace the function and its tangent line on a graph. This will enable you to visualize