Problems

01

Derivative functions - limit definition

Compute the derivative $f^{\prime }(x)$ for each of these functions using the limit definition:

• (a) $f(x)=3x-7$
• (b) $f(x)=1-x^{-1}$
• (c) $f(x)=x-\sqrt{x}$
• (d) $f(x)=\frac{1}{\sqrt{x}}$

02

Derivative functions - derivative rules

Compute the derivative $\frac{df}{dx}$ for each of these functions using the derivative rules.

• (a) $f(x)=x^{10}+\frac{7}{9}{x}^9+\frac{1}{2}{x}^8+\pi x^5-\sqrt{2}{x}^4+e^3$
• (b) $f(x)=3e^x+\frac{5}{x^3}$
• (c) $f(x)=(3x-1)(x+2)$
• (d) $f(x)=\frac{x^2+4x+3}{\sqrt{x}}$

03

Tangent line geometry

Find all points $(x,y)$ on the curve satisfying $y=2x^3+3x^2-12x+5$ where the tangent line to the curve at that point is horizontal.

04

Tangent and normal

• (a) Find the equation of the tangent line to the curve $y=x-\sqrt{x}$ at $x=1$.
• (b) Find the equation of the normal line to the curve $y=x-\sqrt{x}$ at $x=1$.

Note: the normal line is the line perpendicular to the tangent line. If a line has slope $m$, the perpendicular has slope $-1/m$.

05

Determining a polynomial

Find a quadratic polynomial $f(x)$ with the properties that $f(2)=13$ and $f^{\prime }(2)=10$ and $f^{\prime \prime }(2)=6$.

Note: here $f^{\prime \prime }(x)$ is the second derivative, i.e. the derivative of the derivative, or $(f^{\prime }(x){)}^{\prime }$.