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B1.2 & B1.3 – Derivatives of Exponential and Logarithmic Functions

B1.2 & B1.3 – Derivatives of Exponential and Logarithmic Functions. IB Math HL/SL - Santowski. Consider the graph of f(x) = a x and then predict what the derivative graph should look like. (A) Derivatives of Exponential Functions – Graphic Perspective.

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B1.2 & B1.3 – Derivatives of Exponential and Logarithmic Functions

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  1. B1.2 & B1.3 – Derivatives of Exponential and Logarithmic Functions IB Math HL/SL - Santowski

  2. Consider the graph of f(x) = ax and then predict what the derivative graph should look like (A) Derivatives of Exponential Functions – Graphic Perspective

  3. Our exponential fcn is constantly increasing, it is concave up and has no max/min points So our derivative graph should be positive, increasing and have no x-intercepts So then our derivative graph should look very similar to another exponential fcn!! (A) Derivatives of Exponential Functions – Graphic Perspective

  4. So when we use technology to graph an exponential function and its derivative, we see that our prediction is correct Now let’s verify this graphic predication algebraically (A) Derivatives of Exponential Functions – Graphic Perspective

  5. Let’s go back to the limit calculations to find the derivative function for f(x) = bx So we see that the derivative is in fact another exponential function (as seen by the bx equation) which is simply being multiplied by some constant (which is given by the limit expression) But what is the value of the limit?? So then, the derivative of an exponential function is proportional to the function itself (B) Derivatives of Exponential Functions – Algebraic Perspective

  6. Investigate lim h0 (2h – 1)/h numerically with a table of values x y -0.00010 0.69312 -0.00007 0.69313 -0.00003 0.69314 0.00000 undefined 0.00003 0.69316 0.00007 0.69316 0.00010 0.69317 And we see the value of 0.693 as an approximation of the limit Investigate lim h0 (3h – 1)/h numerically with a table of values x y -0.00010 1.09855 -0.00007 1.09857 -0.00003 1.09859 0.00000 undefined 0.00003 1.09863 0.00007 1.09865 0.00010 1.09867 And we see the value of 1.0986 as an approximation of the limit (C) Investigating the Limits

  7. Investigate lim h0 (4h – 1)/h numerically with a table of values x y -0.00010 1.38620 -0.00007 1.38623 -0.00003 1.38626 0.00000 undefined 0.00003 1.38633 0.00007 1.38636 0.00010 1.38639 And we see the value of 1.386 as an approximation of the limit Investigate lim h0 (eh – 1)/h numerically with a table of values x y -0.00010 0.99995 -0.00007 0.99997 -0.00003 0.99998 0.00000 undefined 0.00003 1.00002 0.00007 1.00003 0.00010 1.00005 And we see the value of 1.000 as an approximation of the limit (C) Investigating the Limits

  8. (D) Special Limits - Summary • The number 0.693 (coming from our exponential base 2), 1.0896 (coming from base = 3), 1.386 (base 4) are, as it turns out, special numbers  each is the natural logarithm of its base • i.e. ln(2) = 0.693 • i.e. ln(3) = 1.0896 • i.e. ln(4) = 1.386

  9. (E) Derivatives of Exponential Functions - Summary • The derivative of an exponential function was • Which we will now rewrite as • And we will see one special derivative  when the exponential base is e, then the derivative becomes:

  10. (F) Examples • Find the equation of the line normal to f(x) = x2ex at x = 1 • Find the absolute maximum value of f(x) = xe-x • Where is f(x) = e –x^2 increasing?

  11. Consider the graph of f(x) = logax and then predict what the derivative graph should look like (G) Derivatives of Logarithmic Functions – Graphic Perspective

  12. Our log fcn is constantly increasing, it is concave down and has no max/min points So our derivative graph should be positive, decreasing and have no x-intercepts (G) Derivatives of Logarithmic Functions – Graphic Perspective

  13. So when we use technology to graph a logarithmic function and its derivative, we see that our prediction is correct Now let’s verify this graphic predication algebraically (G) Derivatives of Logarithmic Functions – Graphic Perspective

  14. (H) Derivatives of Logarithmic Functions – Algebraic Perspective • Let logbx = y  so then by = x • So now we have an exponential equation (for which we know the logarithm), so we simply use implicit differentiation to find dy/dx • d/dx (by) = d/dx (x) • [ln(b)] x by x dy/dx = 1 • dy/dx = 1/[byln(b)]  but recall that by = x • Dy/dx = 1/[x  ln(b)] • And in the special case where b = e (i.e. we have ln(x)), the derivative is 1/[x  ln(e)] = 1/x

  15. (I) Derivatives of Logarithmic Functions - Summary • The derivative of a logarithmic function is • And we will see one special derivative  when the exponential base is e, then the derivative of f(x) = ln(x) becomes

  16. (J) Examples • Find the maximum value of f(x) = [ln(x)] ÷ x • Find f `(x) if f(x) = log10(3x + 1)10 • Find where the function y = ln(x2 – 1) is increasing and decreasing • Find the equation of the tangent line to y = ln(2x – 1) at x = 1

  17. (K) Internet Links • Calculus I (Math 2413) - Derivatives - Derivatives of Exponential and Logarithm Functions from Paul Dawkins • Visual Calculus - Derivative of Exponential Function • From pkving

  18. (L) Homework • Stewart, 1989, Chap 8.2, p366, Q4-10 • Stewart, 1989, Chap 8.4, p384, Q1-7

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