The tangent to #y=x^2e^x# at #x=1# cuts the #x# and #y#-axes at #A# and #B# respectively. Find the coordinates of #A# and #B#.?

#y=x^2e^x#

Answer 1

See below

For the slope of the tangent,

#dy/dx = x^2*e^x+e^x*2(x)=x^2*e^x+2e^x x#
the slope of the tangent at #x=1#
#dy/dx = 3e#
Putting #x=1# in the given equation,
#y = x^2*e^x#
#y = e#
So, the tangent passes through the point (1,e) And, it has a slope of #dy/dx = 3e#

So the equation of the tangent is given by,

#y = mx +c#

where m is the slope.

Substituting the values,

#y = 3ex + c#

It also passes through (1,e)

So,

#e = 3e + c#
#c = -2e#

therefore the equation of the tangent is,

#y = 3ex -2e#
For point A, put #y=0#
#x=2/3#

Therefore point A is,

#A(2/3,0)#
For point B, put #x=0#
#y = -2e#

Therefore point B is,

#B(0,-2e)#
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Answer 2

# A=(2/3,0)# and #B=(0,-2e) #

We have a curve given by the equation:

# y=x^2e^x #

First we note that when #x=1#, we have:

# y = 1e^1 =e #

So the tangent passes through #P(1,e)#

The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point. So if we differentiate the equation we have:

# dy/dx = (x^2)(d/dxe^x) + (d/dxx^2)(e^x) #
# \ \ \ \ \ \ = x^2e^x + 2xe^x #
# \ \ \ \ \ \ = (x^2 + 2x)e^x #

And so the gradient of the tangent at #x=1# is given by:

# m = [dy/dx]_(x=1) #
# \ \ = (1+2)e^1 #
# \ \ = 3e #

So, using the point/slope form #y-y_1=m(x-x_1)# the tangent equations is;

# y - e = 3e(x-1) #
# :. y - e = 3ex-3e) #
# :. y = 3ex-2e #

At #A# we have #y=0#:

# :. 0 = 3ex-2e => x=2/3 #

At #B# we have #x=0#:

# :. y = -2e #

Hence the required coordinates are:

# A=(2/3,0)# and #B=(0,-2e) #

(Rounding #e# to 2dp then #P=(1,5.44)# and the tangent equation is #y=8.15x-5.44# making #A=(0.67,0)# and #B=(-5.44,0)#)

We can verify this solution graphically:

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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