Limits
a) Give a formal definition of and explain in your own words and using a picture the meaning of and lim x àa f(x) =L,
lim x àa+ f(x) =L, lim x àa- f(x) =L, where a is a finite number and L can be either a finite number or (positive or negative) infinity.
b) Evaluate the above limits either graphically or algebraically, or determine the limit does not exist; if the limit is of the form 0/0 use appropriate techniques to evaluate it.
c) Evaluate limits that contain a parameter.
d) State and apply the Squeeze theorem.
e) Give a formal definition of continuity and explain in your own words what it means for a function to be continuous at a point and on an interval.
f) Give examples of continuous and discontinuous functions that model real-life quantities.
g) Determine the parameters of a piecewise function that make the function continuous.
h) State the Intermediate Value Theorem, and recognize when it is applicable.
i) Use the Intermediate Value Theorem to estimate roots of equations.
Derivatives and Antiderivatives
a) Using difference quotients, explain how to define the instantaneous rate of change of a function/slope of tangent line as a limit, and argue why these concepts require the notion of limit.
b) Give a formal definition of derivative as a limit, both at a point and as a function.
c) Give a definition of higher order derivatives, and their interpretation in terms of rates of change.
d) Using the limit definition, calculate derivatives of polynomial functions of degree 3 or less, of rational functions p(x)/q(x) where p and q are polynomials of degree 3 or less, and of simple square root functions.
e) Given the equation of a curve of equation y = f(x),
i) relate the derivative f ’ to the slope of a line tangent to the curve,
ii) compute the slope of a tangent at a given point,
iii) find the equation of lines tangent to the curve,
j) solve geometrical problems involving tangent lines.
f) Given the curve of a position function,
i) relate the slope of a tangent line to the value of instantaneous velocity, and vice versa, and compute instantaneous velocities using derivatives,
ii) give a mathematical interpretation of instantaneous acceleration, and compute instantaneous accelerations using derivatives.
g) Give a mathematical definition of antiderivative, and find antiderivatives of polynomial functions and basic trigonometric functions given some initial conditions.
h) Given the graph of a function f, sketch the graph of f ’ and f ’’, and vice versa.
i) Given graphs of position, velocity and acceleration of a particle, identify which is which.
j) Give a geometrical proof for why lim x à0 sin(x)/x =1, and use this limit to prove that (sin x)’ = cos x.
k) Give an empirical explanation for why lim x à0 (ex – 1)/x = 1, and use it to define the derivative of ex, and extend the idea to any base.
l) Apply appropriate differentiation rules (individually or in combination with other rules) to compute derivatives analytically.
k) Using either the definition of derivative as a limit or appropriate differentiation rules, prove simple statements or equations involving derivatives.
a) Define what it means for a function to be differentiable.
b) Explain why differentiable functions are continuous.
c) Demonstrate, using an example, that continuous functions need not be differentiable.
d) Determine the parameters of a piecewise function that make the function differentiable.
Implicit differentiation and logarithmic differentiation
a) Given an implicit relationship between two variables x and y, find dy/dx by applying the technique of implicit differentiation.
b) Justify the technique of implicit differentiation using the Chain Rule.
c) Given a curve f(x,y) = g(x,y) composed of simple functions, find the equation of lines tangent to the curve.
d) Use implicit differentiation to find the derivative of arcsin(x) and arccos(x) and prove that (ln x)’ =1/x.
e) Use logarithmic differentiation to differentiate products of functions, and functions of the form [f(x)]g(x) .
Applications of the Derivative
Linear Approximation:
e) Explain using a graph how to obtain the linear approximation of a function at a point
f) Compute linear approximations of a given function
g) Use a linear approximation to estimate function values
Rates problems:
a) Use derivatives to compute rates of change
a) solve related rates problems, that is, problems where two or more quantities are related to each other and one seeks the instantaneous rate of change of one quantity given the rate of change of one of the other ones.