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

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

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

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

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

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

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

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 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(2) = 0.693 i.e. ln(3) = 1.0896 i.e. ln(3) = 1.0896 i.e. ln(4) = 1.386 i.e. ln(4) = 1.386

9. (E) Derivatives of Exponential Functions - Summary The derivative of an exponential function was The derivative of an exponential function was Which we will now rewrite as Which we will now rewrite as And we will see one special derivative  when the exponential base is e, then the derivative becomes: 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) = x 2 e x at x = 1 Find the equation of the line normal to f(x) = x 2 e x at x = 1 Find the absolute maximum value of f(x) = xe -x Find the absolute maximum value of f(x) = xe -x Where is f(x) = e –x^2 increasing? Where is f(x) = e –x^2 increasing?

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

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

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

14. (H) Derivatives of Logarithmic Functions – Algebraic Perspective Let log b x = y  so then b y = x Let log b x = y  so then b y = x So now we have an exponential equation (for which we know the logarithm), so we simply use implicit differentiation to find dy/dx 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 (b y ) = d/dx (x) d/dx (b y ) = d/dx (x) [ln(b)] x b y x dy/dx = 1 [ln(b)] x b y x dy/dx = 1 dy/dx = 1/[b y  ln(b)]  but recall that b y = x dy/dx = 1/[b y  ln(b)]  but recall that b y = x Dy/dx = 1/[x  ln(b)] 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 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 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 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 the maximum value of f(x) = [ln(x)] ÷ x Find f `(x) if f(x) = log 10 (3x + 1) 10 Find f `(x) if f(x) = log 10 (3x + 1) 10 Find where the function y = ln(x 2 – 1) is increasing and decreasing Find where the function y = ln(x 2 – 1) is increasing and decreasing Find the equation of the tangent line to y = ln(2x – 1) at x = 1 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 Calculus I (Math 2413) - Derivatives - Derivatives of Exponential and Logarithm Functions from Paul Dawkins Calculus I (Math 2413) - Derivatives - Derivatives of Exponential and Logarithm Functions from Paul Dawkins Calculus I (Math 2413) - Derivatives - Derivatives of Exponential and Logarithm Functions from Paul Dawkins Visual Calculus - Derivative of Exponential Function Visual Calculus - Derivative of Exponential Function Visual Calculus - Derivative of Exponential Function Visual Calculus - Derivative of Exponential Function From pkving From pkving From pkving From pkving

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