How do you use the limit definition to find the slope of the tangent line to the graph #f(t) = t − 13 t^2# at t=3?

Answer 1

# f'(3)=-77#

The slope of the tangent at any point is given by the derivative of the function at that point. The definition of the derivative of #y=f(x)# is
# f'(x)=lim_(h rarr 0) ( f(x+h)-f(x) ) / h #
So with # f(t) = t-13t^2 # and the value of #f'(3)# sought then;
# f'(3)=lim_(h rarr 0) (f(3+h)-f(3))/h# # \ \ \ \ \ \ \ \ \=lim_(h rarr 0) ({(3+h)-13(3+h)^2}-{3-13*3^2})/h#
# \ \ \ \ \ \ \ \ \=lim_(h rarr 0) ({3+h-13(9+6h+h^2)}-{3-13*9})/h#
# \ \ \ \ \ \ \ \ \=lim_(h rarr 0) ({3+h-117-78h-13h^2}-{3-117})/h#
# \ \ \ \ \ \ \ \ \=lim_(h rarr 0) {3+h-117-78h-13h^2-3+117)/h#
# \ \ \ \ \ \ \ \ \=lim_(h rarr 0) {h-78h-13h^2)/h#
# \ \ \ \ \ \ \ \ \=lim_(h rarr 0) (-77-13h)#
# \ \ \ \ \ \ \ \ \=-77#
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Answer 2

To find the slope of the tangent line to the graph of f(t) = t - 13t^2 at t=3 using the limit definition, we can follow these steps:

  1. Start with the given function f(t) = t - 13t^2.
  2. Determine the derivative of f(t) with respect to t, denoted as f'(t), by applying the power rule and constant rule. In this case, f'(t) = 1 - 26t.
  3. Substitute the value t=3 into the derivative f'(t) to find the slope at that specific point. Thus, f'(3) = 1 - 26(3) = -77.
  4. The slope of the tangent line to the graph of f(t) at t=3 is -77.
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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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