**concavity**will tell you whether your estimate is an underestimate or an overestimate. You can see this if you draw any concave up curve, then draw a tangent line somewhere to this curve and see if the line is above (overestimate) or below (underestimate); a concave down curve is similar.

Is your healthy bread good for you?

**is bread bad for you when losing weight**.

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If the graph is increasing on the interval, then **the left-sum is an underestimate of the actual value** and the right-sum is an overestimate. If the curve is decreasing then the right-sums are underestimates and the left-sums are overestimates.

Overestimate means ‘to form too high an estimate of’ (see Oxford Dictionaries). **Underestimate means** to estimate that something is smaller or less important than it actually is (see Oxford Dictionaries).

If f (t) > 0 for all t in I, then f is concave up on I, so L(x0) < f(x0), so your approximation is an under-estimate. **If f (t) < 0 for all t in I, then f is concave down on I, so L(x0) > f(x0)**, so your approximation is an over-estimate.

If f is increasing, then its minimum will always occur on the left side of each interval, and its maximum will always occur on the right side of each interval. So for increasing functions, the left Riemann sum is always an underestimate and **the right Riemann sum is always an overestimate**.

When the estimate is higher than the actual value, it’s called an overestimate. **When the estimate is lower than the actual value**, it’s called an underestimate.

A statistic is biased if, in the long run, it consistently over or underestimates the parameter it is estimating. … A statistic is **positively biased if it tends to overestimate the** parameter; a statistic is negatively biased if it tends to underestimate the parameter.

Function is always decreasing → LEFT is an overestimate, RIGHT is an underestimate. Function is **always concave up** → TRAP is an overestimate, MID is an underestimate.

An underestimate is **an estimate that is less than the actual answer to a problem**.

So the concavity of a function can tell you whether the linear approximation will be an **overestimate** or an underestimate. 1. If f(x) is concave up in some interval around x = c, then L(x) underestimates in this interval.

The midpoint approximation **underestimates for a concave up (aka convex) curve**, and overestimates for one that is concave down. There’s no dependence on whether the function is increasing or decreasing in this regard.

NOTE: The Trapezoidal Rule overestimates a curve that is **concave up and underestimates functions that are concave down**.

We know that if the function is concave down, then the tangent line will be above the function. Hence, using the tangent line as an approximation will give an **overestimated value**.

The midpoint Riemann sum over each small interval is an underestimate if the second derivative is negative, and an overestimate **if the second derivative is positive uniformly over the interval**.

**The left approximation** is an underestimate when f(x) is an increasing function, and an overestimate when f(x) is a decreasing function.

overestimate. • **to estimate a value that is over the exact value**.

Underestimation can **impact dependencies** and the overall quality of the project. Overestimation may be wasteful for the resources on a particular task, but it is less likely to impact other tasks or overall quality.

The expressions **0⋅∞,∞−∞,1∞,∞0, and 00** are all considered indeterminate forms. These expressions are not real numbers. Rather, they represent forms that arise when trying to evaluate certain limits.

1 : to estimate as being less than the actual size, quantity, or number. 2 : to place too low a value on : **underrate**.

to guess **an amount that is too high or a size that is too big**: I overestimated and there was a lot of food left over after the party. Most of us overestimate how much time and energy we have. According to the survey, agents frequently overestimate the size of rooms.

**Sampling variability** refers to how much the estimate varies from sample to sample. Have you ever noticed that some bathroom scales give you very different weights each time you weigh yourself? With this in mind, let’s compare two scales.

A graph is said to be concave up at a point if **the tangent line to the graph at that point lies below the graph in the vicinity of the point** and concave down at a point if the tangent line lies above the graph in the vicinity of the point.

Concavity relates to the rate of change of a function’s derivative. A function **f is concave up (or upwards)** where the derivative f′ is increasing. … Similarly, f is concave down (or downwards) where the derivative f′ is decreasing (or equivalently, f′′f, start superscript, prime, prime, end superscript is negative).

If f′′(a)>0, f ″ ( a ) > 0 , then we know the graph of f is concave up, and we see the first possibility on the left, where the tangent line lies entirely below the curve. **If f′′(a)<0, f ″ ( a ) < 0 , then f is concave down** and the tangent line lies above the curve, as shown in the second figure.

When someone underestimates you, **they are giving you an opportunity**. They have no high expectations of what you can bring to the table, and the element of surprise that you are able to deliver makes people pay attention. Don’t let underestimation silence you.

1 verb If you underestimate something, **you do not realize** how large or great it is or will be. None of us should ever underestimate the degree of difficulty women face in career advancement… Never underestimate what you can learn from a group of like-minded people. V wh.

A critical point of a function of a single real variable, f(x), is a value x0 in the domain of f where it is not differentiable or its derivative is 0 (f ′(x0) = 0). A critical value is **the image under f of a critical point**. … Notice how, for a differentiable function, critical point is the same as stationary point.

The first derivative test is **the process of analyzing functions using their first derivatives in order to find their extremum point**. This involves multiple steps, so we need to unpack this process in a way that helps avoiding harmful omissions or mistakes.

- Find the point we want to zoom in on.
- Calculate the slope at that point using derivatives.
- Write the equation of the tangent line using point-slope form.
- Evaluate our tangent line to estimate another nearby point.

Unlike the trapezoid and midpoint rules, where at least for curves of a given concavity, we can say whether or not the rule gives an overestimate or an underestimate, **we have no such clear result for Simpson’s rule**.

If a function is INCREASING, LRAM **underestimates the actual area** and RRAM overestimates the actual area. If a function is DECREASING, LRAM overestimates the actual area and RRAM underestimates the actual area.

A Riemann sum is an approximation of the area under a curve by dividing it into multiple simple shapes (like rectangles or trapezoids). … In a right Riemann sum, **the height of each rectangle is equal to the value of the function at the right endpoint of its base**.

In general, when a curve is concave down, trapezoidal rule **will underestimate the area**, because when you connect the left and right sides of the trapezoid to the curve, and then connect those two points to form the top of the trapezoid, you’ll be left with a small space above the trapezoid.

Whereas the main advantage of the Trapezoid rule is its rather easy conceptualization and derivation, Simpson’s rule 2 Page 3 approximations usually achieve a given level of **accuracy faster**. Moreover, the derivation of Simpson’s rule is only marginally more difficult.

(13) **The Midpoint rule is always more accurate than the Trapezoid rule**. … For example, make a function which is linear except it has nar- row spikes at the midpoints of the subdivided intervals. Then the approx- imating rectangles for the midpoint rule will rise up to the level of the spikes, and be a huge overestimate.

An example with negative dx When using linear approximations, **x doesn’t have to be bigger than a**.

The second derivative is **the rate of change of the rate of change of a point at a graph** (the “slope of the slope” if you will). This can be used to find the acceleration of an object (velocity is given by first derivative).

Since the new shape and the original midpoint sum rectangle have the same area, the midpoint sum is also an underestimate for **the area of R.** f(x) = 17 – x2 and the x-axis on the interval [0, 4].