The linear approximation (or tangent line approximation or linearization) of a function at is the tangent line at . In formulas, is the linear function
Notice that this is obtained by writing the tangent line to at in point-slope form and calling the resulting linear function . The linear approximation is a linear function that looks like when we zoom in near .
Compute the second derivative of . What do you notice about the sign of the second derivative of ? What does this tell you about the shape of the graph?
Conclude that because the graph of has a certain shape, the graph will bend below the tangent line and so that will always be smaller than the tangent line approximation .
Someone claims that the square root of 1.1 is about 1.05. Use the linear approximation to check this estimate. Do you think this estimate is about right? Why or why not?
If a function is concave up around , then the function is turning upwards from its tangent line. So when we use a linear approximation, the value of the approximation will be below the actual value of the function and the approximation is an underestimate. If a function is concave down around , then the function is turning downwards from its tangent line. So when we use a linear approximation, the value of the approximation will be above the actual value of the function and the approximation is an overestimate.
Suppose has a continuous positive second derivative and is a small increment in (like in the limit definition of the derivative). Which one is larger...
Find a solution to this system of linear equations! Your answer will give you values of that can be used to draw a quadratic approximating the natural logarithm. You can check your answer on Desmos https://www.desmos.com/calculator/bad2xrwmvl