How do you use the limit definition to find the slope of the tangent line to the graph #y = x^(3) + 2x^(2) - 3x + 2# at x=1?

Answer 1

See below.

We could use #lim_(hrarr0) (f(1+h)-f(1))/h# or #lim_(hrarr0) (f(x+h)-f(x))/h# then substitute #1# for #x#,

but I don't really want to get into cubing a binomial. So I'll use

#lim_(xrarr1)(f(x)-f(1))/(x-1) = lim_(xrarr1) ((x^3+2x^2-3x+2)-(2))/(x-1)#
# = lim_(xrarr1) (x^3+2x^2-3x)/(x-1)#
# = lim_(xrarr1)(x(x^2+2x-3))/(x-1)#
# = lim_(xrarr1)(x(x+3)(x-1))/(x-1)#
# = lim_(xrarr1)x(x+3)#
# = 4#
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Answer 2

To find the slope of the tangent line to the graph y = x^3 + 2x^2 - 3x + 2 at x = 1 using the limit definition, we can follow these steps:

  1. Start with the given function: y = x^3 + 2x^2 - 3x + 2.

  2. Calculate the difference quotient, which represents the slope of the secant line between two points on the graph. Let's choose a point close to x = 1, such as x = 1 + h, where h is a small positive number.

  3. Substitute the chosen point into the function to find the corresponding y-value: y = (1 + h)^3 + 2(1 + h)^2 - 3(1 + h) + 2.

  4. Expand and simplify the expression obtained in step 3.

  5. Subtract the original function evaluated at x = 1 from the expression obtained in step 4.

  6. Divide the result from step 5 by h.

  7. Take the limit as h approaches 0 of the expression obtained in step 6.

The limit obtained in step 7 will give us the slope of the tangent line to the graph at x = 1.

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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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