Given the quadratic equation *y = 3x ^{2} + 2* and the math function

**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.

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*.

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

your answer and the program answer are equal, which means that you are right.

Given the equation of a curve *f(x) = 2x ^{3} - 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

you check graphically whether your answer makes sense or not.

A particle is moving along a straight line *X* according to the following

equation : *x(t) = 2t ^{2} + 8t + 9*, where

the distance of the particle at time

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.

A particle moves along a straight line. Its position, *x*, as a function of time

is given by the equation *x(t) = 2t ^{3} - 12t^{2} + 20t + 3* where

the distance (in meters) traveled at time

**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*

Find the coordinates of the points on the curve *y = x ^{3} - 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

your equations and to check whether your answers are correct or not.

Given the curve described by the following function *y = (1/2)x ^{2} - 2x + 3*, at which

point of the graph the tangent line to the curve is perpendicular to another tangent

line that passes through the point

When you will have figured out the answer, use the free online Graph Plotter to

trace the curve and its tangent lines on a graph with an XY coordinate grid. This will

help you to visualize your equations and it will enable you to check whether or not

your answers are correct.