Exponential and trigonometric functions practice question 18
From the definition of the derivative, for every base , where . A calculator estimates . Without estimating another limit, find the slope of at .
The slope at is , so everything rests on . Since , how is related to ?
Factor a difference of squares: . Divide by and let .
As , , so . Multiply by the height of the graph at .
A decimal to three decimal places.
Theory UBC Math 100 §3.5Key results
Worked solution
Strategy
The derivative of an exponential is its own height times a constant that depends only on the base. Relate the constant for base 9 to the one for base 3 by factoring inside the limit, then multiply by the height of 9x at x = 1/2.
Working
9h = (3h)², so 9h − 1 is a difference of squares.
Both limits exist, and 3h → 30 = 1 as h → 0, so the second factor tends to 2.
The slope of an exponential is the constant for its base times its height, and 91/2 = 3.
With the calculator’s estimate of C(3). Section 4.4 of the course shows C(a) = log a, and log 9 = 2 log 3 agrees with C(9) = 2C(3).
Answer
The answer is 6 times C(3), about 6.592.
Check
The secant slopes of 9x just either side of x = 1/2, over steps of 0.01, are about 6.52 and 6.66, and 6.592 sits between them.
Where this goes wrong
Squaring C(3) because the base is squared. The factorisation 9h − 1 = (3h − 1)(3h + 1) shows the constant only doubles, since the second factor tends to 2.
The idea behind it
The exponential and trigonometric functions lesson, a figure to move, in a few steps
Leans on
The derivative of ex from the definition, from The definition of the derivative
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