Natural Exponential Function

4. Problem-Solving Techniques

• Simplifying Natural Exponential Expressions: Making Good Use of Exponential Rules
  • Keep in mind the core rules of exponential operations, including multiplication ( $e^m \cdot e^n = e^{m + n}$ ), division ( $e^m \div e^n = e^{m - n}$ ), and power of a power ( $(e^m)^n = e^{mn}$ ). Ensure that the rules are not confused during simplification, and especially note that the base is always e, so there is no need for additional base conversion.
• Analyzing Function Transformations: "Step-by-Step Decomposition + Key Points Migration"
  • When dealing with a transformed natural exponential function (such as  $y = a e^{x - h} + k$ ), decompose the transformation steps in the order of "reflection/stretching → horizontal translation → vertical translation" to avoid step confusion;
  • First, find the three key points of the standard function  $y = e^x$  ( $(-1, \frac{1}{e})$ ,  $(0, 1)$ ,  $(1, e)$ ), then adjust the coordinates according to each transformation rule (e.g., subtract 2 from x for a 2-unit left shift, multiply y by 3 for a 3-fold stretch), and finally connect the new points and mark the horizontal asymptote to quickly draw the graph.
• Solving Continuously Compounded Interest Problems: "Four-Step" Formula Application
  • Step 1: "Identify Parameters" — Extract the principal P, annual interest rate r (must be converted to a decimal), and time t from the problem. Clarify the unit of each parameter (e.g., t must be consistent with the time unit of r, both in "years");
  • Step 2: "Substitute into the Formula" — Substitute the parameters into the continuously compounded interest formula  $A(t) = P e^{rt}$ , and ensure the correct calculation of the exponential part (no calculation error in the product of  $r \times t$ );
  • Step 3: "Calculate the Exponent" — Use a calculator or the approximate value of e ( $e \approx 2.718$ ) to calculate  $e^{rt}$ . If the exponent is a decimal (e.g., 0.2), obtain the accurate approximate value through a table lookup or calculator;
  • Step 4: "Find the Result" — Calculate  $P \times e^{rt}$ , and round the result according to the requirements of the problem (e.g., rounding to the nearest whole number, keeping two decimal places).
• Verifying Results: Checking with Function Properties
  • For natural exponential function graph problems, verify whether the key points conform to the transformation rules (e.g., whether the coordinates are correct after translation) and whether the horizontal asymptote changes with vertical translation;
  • For continuously compounded interest problems, verify whether the result is reasonable: Since the interest from continuous compounding is higher than that from ordinary compounding, the result should be slightly higher than the amount calculated with annual compounding ( $A = P(1 + r)^t$ ) (for example, in this problem, the amount calculated with annual compounding is  $3000 \times (1 + 0.05)^4 \approx 3646$ , and the continuous compounding result of 3664 is slightly higher, which is in line with expectations). If the result is obviously abnormal, check the parameter substitution or calculation process.