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Suppose that we don't have a formula for $ g(x) $ but we know that $ g(2) = -4 $ and $ g'(x) = \sqrt {x^2 + 5} $ for all $ x. $(a) Use a linear approximation to estimate $ g(1.95) $ and $ g(2.05). $(b) Are your estimate in part (a) too large or too small? Explain.
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Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 10
Linear Approximation and Differentials
Derivatives
Differentiation
Oregon State University
Harvey Mudd College
University of Michigan - Ann Arbor
Idaho State University
Lectures
04:40
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
44:57
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
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Okay. The first thing you know can do is we could substitute G prime of two into two squared plus five, which is a square to four plus five, which is scared of mine, which is equivalent to three. So now we have a G F ax is negative for plus three times X minus two, which is negative for post three acts minus six, which is three X minus 10. Therefore, we have G of 1.95 this three times 1.95 I was 10 negative. 4.15 Three times 2.5 minus tongue as negative. 3.85 Okay, we know that the second derivative is gonna be axe over the square root of ax squared. Plus five. Don't fit the chain role. No, we know the linear approximation is gonna be a line there for all the points away from the middle point to negative four from the center. X equals two are less than the actual values. Remember, G is calm cave up, so the estimates are less. Therefore, we haven't underestimate
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