Q1. Use the identity a=e^logea, to express the following as a power of e, and hence find its primitive.
(a) 2^x (b) 5^-x
Q2. Find a primitive of: (6x^2 - 8x + 6x).e^(x^3 -2x^2 +3x -5)
Q3. How do you sketch a primitive function given a picture of a graph? Help
In Question 1, start with \(2=e^{\log_e{\left(2\right)}}\Rightarrow 2^x=\left(e^{\log_e{\left(2\right)}}\right)^x=e^{x\log_e{\left(2\right)}}\) then use the rule for the primitive of an exponential, \(\int{e^{ax}}dx=\frac{1}{a}e^{ax}+c,c\in\mathbb{R}.\)
For Question 2, note that \(\frac{d}{dx}\left(e^{x^3-2x^2+3x-5}\right)=\left(3x^2-4x+3\right)e^{x^3-2x^2+3x-5}=\frac{1}{2}\left(6x^2-8x+6\right)e^{x^3-2x^2+3x-5}\)
For your final question, it is often easier to try and think in reverse (i.e. the graph given looks as it does due to the fact that it is the graph of the derivative of the graph that you are trying to draw).
This means that:
- when the given graph is at zero (i.e. an x-intercept), the gradient of the primitive graph is zero (so a stationary point at the same x-value)
- when the given graph is greater than zero (i.e. above the x-axis), the gradient of the primitive graph is positive at the same x-values
- when the given graph is less than zero (i.e. below the x-axis), the gradient of the primitive graph is negative at the same x-values
- when the given graph is increasing, the gradient of the primitive graph is increasing
- when the given graph is decreasing, the gradient of the primitive graph is decreasing
Due to each of these properties, the y-values of the primitive graph can't be explicitly stated, unless more information is given (there are infinitely many possibilities, each of which is a vertical translation of every other)
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