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Understanding Derivatives in Calculus: Rates of Change and Tangents

Learn about the derivative - a function measuring change rates, tangents, slopes, secants, and more in calculus. Explore examples and practice problems to deepen your understanding.

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Understanding Derivatives in Calculus: Rates of Change and Tangents

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  1. The Derivative Eric Hoffman Calculus PLHS Sept. 2007

  2. Key Topics • Derivative : a new function that we can use to measure rates of change for a given function • Rate of Change : the rate at which the y-coordinate changes with respect to the x-coordinate • Derivative of a straight line: the derivative of a straight line is just the slope of the line

  3. Key Topics • Tangent: the line tangent to the curve C at point P is the straight line that most resembles the curve at point P • Look at example on pg. 80 of your book • Tangent is important because it is a straight line and we can calculate the slope of that line at point P to determine the slope of curve C at point P.

  4. Key Topics • If we assume the graph of a curve is a function we choose points P and Q so that they are on the function • Secant : the line that passes through the points P and Q. Note: As point Q gets closer and closer to point P it gives us a closer and closer approximation for the tangent line at point P. Example 1: Click Here

  5. Key Topics • Slope of a secant line : the slope of a secant line through two points on a curve is: Change in y-values Change in x-values

  6. Understanding Slope of a Secant line Note: Slope = works the same as Also note: y1 is the same thing as f(x1), y0= f (x0), etc. • if we let h = (x1-x0), a fancy way of writing the difference between two points on the x-axis • If we let x1 = x0+ h i.e. thefirst pointplus the difference • Knowing all this, by using the substitution property we can say: Slope

  7. Key Topics • If l is the line tangent to the graph of y = f(x) at the point (x0 , f (x0)) then the slope m of l is: • Read as “the limit as h approaches zero of …” • Basically means allowing the h to shrink to zero For Example:

  8. Key Topics • Look at example 1 on page 84 in your book: Make sure you read the strategy located at the left hand side of the page. This will help you understand each step involved. Now, using the same approach, at your desk calculate the slope of the line tangent to f (x) = x3 + x at the point (2 , 10)

  9. line tangent to f (x) = x3 + x at the point (2 , 10) Step 1: Identify x0 : x0 = 2 Step 2: Substitute for f (x0+h) and f (x0) Step 3: Simplify using properties of Algebra Step 4: Let h approach 0 and evaluate the limit

  10. Graph of Tangent to f(x)=x3 + x at (2,10)

  11. Key Topics • Homework 2.1 pg. 90 #1-24 all and #28 • Work in pairs to try and teach each other. • Homework will be collected on Friday • If you have any questions, read section 2.1 in your textbook, there are lots of examples

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