5.1 Limits and the derivativeIB Maths: Analysis and Approaches HL: Flashcards

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What does $\lim_{h\to0}(5+h)$ equal?

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What does $\lim_{h\to0}(5+h)$ equal?: 5
What is a limit, informally?: The value an expression gets closer and closer to as its input approaches a given value.
Write the gradient of the chord from $x$ to $x+h$ on $y=f(x)$ .: $\frac{f(x+h)-f(x)}{h}$
How is the gradient of a curve at a point found as a limit?: It is the limit of the gradients of chords $PQ$ as $Q$ approaches $P$ ( $h\to0$ ).
What two interpretations does the derivative have?: The gradient function of a curve, and the instantaneous rate of change of one quantity with respect to another.
What does $\frac{\mathrm{d}V}{\mathrm{d}r}$ mean?: The rate of change of $V$ with respect to $r$ .
Give two notations for the derivative of $y=f(x)$ .: $\frac{\mathrm{d}y}{\mathrm{d}x}$ and $f'(x)$ .
What are the units of $\frac{\mathrm{d}s}{\mathrm{d}t}$ if $s$ is in metres and $t$ in seconds?: m s $^{-1}$ (metres per second).
What does a negative derivative tell you?: The quantity is decreasing at that point.
Average versus instantaneous rate of change?: Average is the gradient of a chord over an interval; instantaneous is the gradient of the tangent at a point, the limit of average rates.
Chord gradients for $y=x^{2}-2x$ at $x=3$ are $4.1, 4.01, 4.001$ . What is the gradient of the curve at $x=3$ ?: 4
Can a limit exist where the expression is undefined?: Yes: $\frac{4h+h^{2}}{h}$ is undefined at $h=0$ but its limit as $h\to0$ is 4.
Are formal analytic methods for limits required at SL?: No, only an informal understanding and estimation from values or graphs.