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Suppose that the only information we have about a function $ f $ is that $ f(1) = 5 $ and the graph of its $ derivative $ is as shown.(a) Use a linear approximation to estimate $ f(0.9) $ and $ f(1.1). $(b) Are your estimates 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
Baylor University
Idaho State University
Boston College
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, First things first. We know we have after 0.9 is off of one minus zero point one, which is half of one class after prime of one times negative 0.1, which is five minus two time 0.1, which is 4.8. So now we have half of 1.1 is half of one plus off. Prime of one come 0.1 is five plus chu time 0.1, which is 5.2. Okay, So part B, as we can see after prime of access positive and decreasing there for the slopes of the tangent lines are positive and are becoming less steep. Therefore, we know that because the tangent line is above the curve of f of axe, we know the estimates must be too big. So it's an overestimate above the curve.
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