2. Intro to Derivatives

Equations of tangent lines by definition (2)

b. Determine an equation of the tangent line at P.

f(x) = √x+3; P (1,2)

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = x⁴

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = (1/x) - x²

Find the derivative function f' for the following functions f.

f(x) = √3x+1; a=8

Find the derivative function f' for the following functions f.

f(x) = 2/3x+1; a= -1

Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.

f(x) = 1/x; a= -5

Consider the line f(x)=mx+b, where m and b are constants. Show that f′(x)=m for all x. Interpret this result.

Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.

Use the definition of the derivative to determine d/dx (√ax+b), where a and b are constants.

A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.

y=3x−4; P(1, −1)

A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.

y= √x; P(4, 2)

A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.

y = 2/x; P(1, 2)

Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.

f(x)=x²+1; Q(3, 6)

Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.

f(x) = 1/x; Q (-2, 4)

If f′(x)=3x+2, find the slope of the line tangent to the curve y=f(x) at x=1, 2, and 3.