Question

Derivatives from limits The following limits represent $f^{\prime }(a)$ for some function $f$ and some real number $a$ a. Find a possible function $f$ and number $a$. b. Evaluate the limit by computing $f^{\prime }(a)$. $$\underset{x\to 1}{lim}\frac{x^{100}-1}{x-1}$$

Short answer

Answer: The given limit represents the derivative of the function $f(x)=x^{100}$ at the point $a=1$. The value of the limit is 100.

Step-by-step solution

Recognize the derivative pattern in the limit

The given limit has the form: $$\underset{x\to a}{lim}\frac{f(x)-f(a)}{x-a}$$ which resembles the definition of the derivative. Let's compare this with the given limit: $$\underset{x\to 1}{lim}\frac{x^{100}-1}{x-1}$$

Identify the function $f$ and the point $a$

Comparing these two expressions, we can identify the function $f$ and the point $a$: $$f(x)=x^{100},a=1$$

Find the derivative $f^{\prime }(x)$

To find the derivative $f^{\prime }(x)$, we can use the power rule: $$f^{\prime }(x)=\frac{d}{dx}(x^{100})=100x^{99}$$

Evaluate the limit by computing $f^{\prime }(a)$

Now, we substitute the value of $a$ into the derivative $f^{\prime }(x)$ to find the limit: $$f^{\prime }(1)=100(1{)}^{99}=100$$ Thus, the given limit evaluates to 100, which represents the derivative of the function $f(x)=x^{100}$ at the point $a=1$.

Key concepts

limit definition of derivative

The limit definition of a derivative is a fundamental concept in calculus. It depicts how a function's behavior can be described in terms of its instantaneous rate of change at a specific point. The formula for this is:$$\underset{x\to a}{lim}\frac{f(x)-f(a)}{x-a}=f^{\prime }(a)$$This equation means that as $x$ approaches $a$, the expression $\frac{f(x)-f(a)}{x-a}$ becomes the slope of the tangent line to the curve $y=f(x)$ at $x=a$. This captures the idea of a derivative as the "instantaneous" rate of change rather than an average rate of change. Understanding this limit form is crucial because it forms the base for calculating derivatives, providing a way to rigorously define them.

power rule for derivatives

The power rule is a straightforward yet powerful tool for finding derivatives quickly, without reverting to limits. It's especially useful for polynomials. The rule states:$$\frac{d}{dx}(x^n)=n{x}^{n-1}$$In essence, the power rule tells us that for any term $x^n$, the derivative is $n{x}^{n-1}$, where $n$ is a real number. Applying this rule simplifies the process of differentiation significantly. When using the power rule, simply multiply the exponent by the coefficient and reduce the exponent by one. This makes it not only efficient but also easily memorized, which is highly beneficial when dealing with high-degree polynomials.

derivative evaluation

Once a derivative is found using a specific rule or method, it is essential to evaluate it at a particular point. This step involves substituting the value into the derivative expression to find the rate of change at that point.For instance, if we have a function $f(x)=x^{100}$ and its derivative is $f^{\prime }(x)=100x^{99}$, to evaluate this derivative at $x=1$, we plug 1 into the derivative function:$$f^{\prime }(1)=100(1{)}^{99}=100$$This result indicates the slope or the rate of change of the original function at that specific point. Evaluating derivatives at a point is crucial in practical scenarios such as physics or engineering, where instantaneous values determine specific conditions or decisions.

derivatives at a point

Derivatives at a point give us the slope of the tangent to a function at that exact location. This concept is key for understanding how functions behave locally. It’s like zooming in on a curve to see exactly how steep it is at a particular spot.Knowing how to compute the derivative at a specific point helps in tracing the curve’s precise behavior at that point. For example, with $f(x)=x^{100}$, using the power rule yields $f^{\prime }(x)=100x^{99}$. Evaluating at $x=1$ shows that $f^{\prime }(1)=100$, indicating a steep curve at that point.Thus, understanding derivatives at a point not only provides mathematical insight but also aids in applications like physics, where it might inform about velocity or acceleration at an instant.