By utilizing the concept of a limit applied to the rate of change of a function, we are able to define the derivative in terms of a limit. This allows us to accurately calculate the slope of a function at any given point. Essentially, it is defined as the limit of (f(x+h)-f(x))/h as h tends towards 0.
| Topic | Problem | Solution |
|---|---|---|
| None | a. Use the definition $m_{\tan }=\lim _{h \righta… | Given the function \(f(x) = x^4\) and the point \(P(2,16)\). |
| None | 1. Let $f(x)$ be the function given by: \[ f(x)=\… | Let's start by finding the values of $f(a)$ and $f(a+h)$ by substituting $a$ and $a+h$ into the fun… |
| None | Find the derivative of the function using the def… | First, we find the derivative of the function using the definition of derivative. |
| None | Question 6 Find the (exact) direction cosines and… | Given the vector \(\vec{v}=\langle 3,3,2\rangle\), we can find the direction cosines and direction … |
| None | 5 2 nd understand (R\&U): Stewart's pp 95-113 The… | The given limit is the definition of the derivative of a function at a point. |
| None | Consider the function $f(x)=3 \sqrt{x}-4$. (a) Si… | \(f(4+h) = 3\sqrt{4+h}-4\) |
| None | Let $f(x)=-x^{2}+9 x$ Find the slope of the secan… | Let's consider the function \(f(x)=-x^{2}+9 x\). |
| None | $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\lef… | First, we need to simplify the expression inside the limit. We can do this by expanding the terms a… |
| None | Use the four-step process to find the slope of th… | Step 1: Calculate \(f(x+h)\), which is \(- (x+h)^{2} + 3(x+h)\). |
| None | Given the function $f(x)=-1-4 x$, express the val… | The problem is asking for the difference quotient of the function \(f(x) = -1 - 4x\). The differenc… |
| None | \( \frac{d f}{d t}=\lim _{h \rightarrow 0} \frac{… | \( \frac{d f}{d t}=\lim _{h \rightarrow 0} \frac{f(t+h)-f(t)}{h} \) |