Limits
- Evaluate: \[ \lim_{x\to -3}(x^2+4x-5) \]
\[ \lim_{x\to -3}(x^2+4x-5) = (-3)^2+4(-3)-5 = 9-12-5 = -8 \]
- Does the following limit exist? (Hint: Expand first) \[ \lim_{x\to 4}\frac{(x-1)^2}{(x-1)} \]
Simplify first: \(\frac{(x-1)^2}{(x-1)}=\frac{(x-1)(x-1)}{(x-1)}=x-1\) for \(x\neq 1\).
\[ \lim_{x\to 4}(x-1) = 3 \]
The limit exists and equals \(3\).
- Does the following limit exist? \[ \lim_{x\to 5}\frac{2}{x-5} \]
Approaching from the right (\(x>5\)): \(x-5\to 0^+\), so \(\frac{2}{x-5}\to +\infty\).
Approaching from the left (\(x<5\)): \(x-5\to 0^-\), so \(\frac{2}{x-5}\to -\infty\).
Since the function approaches different (infinite) behavior from the left and right, the limit does not exist.
- Let \[ g(x)=\begin{cases}2x-3, & x\geq 1\\x-2, & x<1\end{cases} \] Does the following limit exist? \[ \lim_{x\to 1}g(x) \]
Approaching from the right (\(x\geq 1\)): \(g(x)=2x-3 \to 2(1)-3=-1\).
Approaching from the left (\(x<1\)): \(g(x)=x-2 \to 1-2=-1\).
Since the function approaches the same value from both sides, the limit exists and equals \(-1\).
Rules for differentiation
- Find \(f'(x)\) for \(f(x)=x^7\)
\[ f'(x)=7x^6 \]
- Find \(f'(x)\) for \(f(x)=\sqrt[4]{x}\)
\[ f(x)=x^{1/4} \Rightarrow f'(x)=\frac{1}{4}x^{-3/4} \]
- Find \(y'\) for \(y=x^4+3x^2-20\)
\[ y'=4x^3+6x \]
- Find \(\frac{dy}{dx}\) for \(y=4x^4-\frac{7}{x^3}+9\)
\[ \frac{dy}{dx}=16x^3+\frac{21}{x^4} \]
- Find \(k'(t)\) for \(k(t)=4t^{2/3}-5\ln(t)\)
\[ k'(t)=\frac{8}{3}t^{-1/3}-\frac{5}{t} \]
Higher-order and Partial Derivatives
- Find the third derivative of \(K(w)=w^6+5w^4-3w^2+w\)
\[ K'(w)=6w^5+20w^3-6w+1 \] \[ K''(w)=30w^4+60w^2-6 \] \[ K'''(w)=120w^3+120w \]
- Position is given by \(s(t)=t^3-6t^2+9t\). Find the velocity function \(s'(t)\) and the acceleration function \(s''(t)\).
\[ v(t)=s'(t)=3t^2-12t+9 \] \[ a(t)=s''(t)=6t-12 \]
- Find \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) for \(f(x,y)=3xy^2-x^2+5y-2\), then evaluate both at \((2,1)\).
\[ \frac{\partial f}{\partial x}=3y^2-2x, \qquad \frac{\partial f}{\partial y}=6xy+5 \] At \((2,1)\): \(\left.\frac{\partial f}{\partial x}\right|_{(2,1)}=3(1)^2-2(2)=-1\), and \(\left.\frac{\partial f}{\partial y}\right|_{(2,1)}=6(2)(1)+5=17\)
- Find \(\frac{\partial C}{\partial x}\), \(\frac{\partial C}{\partial y}\), and \(\frac{\partial C}{\partial z}\) for \(C(x,y,z)=2x^2z-5y^3+3xy-z^2+10\)
\[ \frac{\partial C}{\partial x}=4xz+3y, \qquad \frac{\partial C}{\partial y}=-15y^2+3x, \qquad \frac{\partial C}{\partial z}=2x^2-2z \]