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c) Finally, draw a large and clear sketch of the graph of f(x) = z² – 2 and illustrate the tangent line at P(1,-1). Give the slope of the tangent line at P and compute the equation of this tangent line.

c) Finally, draw a large and clear sketch of the graph of f(x) = z² – 2 and illustrate the tangent line at P(1,-1). Give the slope of the tangent line at P and compute the equation of this tangent line.

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.2: Introduction To Conics: parabolas
Problem 4ECP: Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.
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Question c.
c) Finally, draw a large and clear sketch of the graph of f(x) = 1² – 2 and illustrate the tangent line at P(1, -1). Give the slope of the tangent line at P and compute the equation of this tangent line. Note: You may want to recall that, in general, the slope of the tangent line at x = a is in fact given by the derivative of f at x = a or the instantaneous rate of change of f at x = a. As discussed in class, the instantaneous rate of change is viewed as the limit as T + a of rates of change 1e-(@ Put differently, given the curve y = f(x), the slope of the tangent line at P(a, f(a) can be interpreted as a limit of slopes of secant lines PQ as Q comes closer and closer to P along the curve, i.e. f(x) – f(a) lim I -- a This limit can also be written as the limit of a difference quotient as f(a+h) – f(a). lim h0 h As such, in our case, since a=1, we deduce that f'(1), which comes from f(x) – f(1) x – 1 = lim h0 f(1+h) – f(1). f'(1) = lim h gives us nothing but the slope of the tangent line at P(1,–1). You may want to note that we have already computed these things in part a), and that you may just use the answer of f'(1) to give the slope of the tangent line at P(1, –1).
Transcribed Image Text:c) Finally, draw a large and clear sketch of the graph of f(x) = 1² – 2 and illustrate the tangent line at P(1, -1). Give the slope of the tangent line at P and compute the equation of this tangent line. Note: You may want to recall that, in general, the slope of the tangent line at x = a is in fact given by the derivative of f at x = a or the instantaneous rate of change of f at x = a. As discussed in class, the instantaneous rate of change is viewed as the limit as T + a of rates of change 1e-(@ Put differently, given the curve y = f(x), the slope of the tangent line at P(a, f(a) can be interpreted as a limit of slopes of secant lines PQ as Q comes closer and closer to P along the curve, i.e. f(x) – f(a) lim I -- a This limit can also be written as the limit of a difference quotient as f(a+h) – f(a). lim h0 h As such, in our case, since a=1, we deduce that f'(1), which comes from f(x) – f(1) x – 1 = lim h0 f(1+h) – f(1). f'(1) = lim h gives us nothing but the slope of the tangent line at P(1,–1). You may want to note that we have already computed these things in part a), and that you may just use the answer of f'(1) to give the slope of the tangent line at P(1, –1).
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