Questions asked by CJ

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• Why is calculus important?
  
• What is the gradient function used for?
  
• Why do we need the gradient function?
  
• What is an infinite limit?
  
• How do you find limits as x approaches infinity?
  
• Is the function #(x^2-6x+9)/(x-3)# continuous?
  
• How can I prove that a function is continuous?
  
• What is the "rate of change" of a function?
  
• Why is it important to know rates of change?
  
• Are there different kinds of rate of change?
  
• What is the slope of a curve?
  
• How do I find the equation for a tangent line without derivatives?
  
• How do you find the equation of a normal line if you know the equation of the tangent line?
  
• If my tangent line at point (4,8) has the equation #y=5x/6 - 9#, what is the equation of the normal line at the same point?
  
• How do I find the derivative of #f(x)=x^3# from first principles?
  
• How do I find the derivative of #x^2 + 7x -4# using first principles?
  
• How do I find the derivative of #x^3 - 2x^2 + x/4 +6# using first principles?
  
• How do I find the derivative of #f(x)=sqrt(x)# using first principles?
  
• How do I find the derivative of #f(x) = sqrt(x+3)# using first principles?
  
• What is the derivative of #x^n#?
  
• How do I find derivatives of radicals like #sqrt(x)#?
  
• How do you find the derivative of #y = f(x) - g(x)#?
  
• What is the quotient rule?
  
• What is the limit #lim_(x->0)sin(x)/x#?
  
• What is the limit #lim_(x->0)(cos(x)-1)/x#?
  
• Is there a way to find the derivative of sin(x) without limits?
  
• How are sin(x), tan(x), and x related graphically?
  
• How can I find the derivative of #y=e^x# from first principles?
  
• How can I find the derivative of #y=c^x# using first principles, where c is an integer?
  
• What is the derivative of #log_e(x)#?
  
• What is the change of base rule for logarithms?
  
• What are the derivatives of the inverse trigonometric functions?
  
• What are the derivatives of exponential functions?
  
• What are the derivatives of logarithmic functions?
  
• What is a stationary point, or critical point, of a function?
  
• What is special about a turning point?
  
• How do I find local maxima and minima of a function?
  
• How can I use derivatives to find acceleration, given a position-time function?
  
• Do all functions have points of inflection?
  
• What is Newton's Method?
  
• How do I evaluate definite integrals?
  
• How do I evaluate indefinite integrals?
  
• What is the antiderivative of a polynomial?
  
• How do you find the antiderivative of #x^2+5x#, if the point (0,5) exists on the graph of the antiderivative?
  
• How do you evaluate the integral #int_0^4x^3+2x^2-8x-1#?
  
• How can you find a function, if you already know the rate of change of the function?
  
• What are the antiderivatives of #sin(x)# and #cos(x)#?
  
• What is the antiderivative of #tan(x)#?
  
• What is the antiderivative of #sec^2(x)#?
  
• What are the antiderivatives of #sec(x)#, #csc(x)# and #cot(x)#?
  
• What is the antiderivative of #e^x#?
  
• What is the antiderivative of #n^x#?
  
• What is a rational function?
  
• How do I find the integral of a rational function?
  
• How do I divide one polynomial by another?
  
• What is the constant of integration and why is it so important?
  
• How do I evaluate constants of integration?
  
• When integrating by trigonometric substitution, what are some useful identities to know?
  
• How does Simpson's Rule work?
  
• Why do we need to approximate integrals when we can work them out by hand?
  
• How do you determine the amount of work needed for movement of objects?
  
• Why is the error of approximation of an integral important?
  
• How does the formula #1/90((b-a)/2)^5(f^(4)(zeta))# work for calculating error?
  
• How do I integrate with Euler's method by hand?
  
• How do I integrate with Euler's method with a calculator or computer?
  
• How do I find the surface area of a solid of revolution using parametric equations?
  
• How are certain formulæ for areas of circles and ellipses related to calculus?
  
• How do you find areas bounded by polar curves using calculus?
  
• How do I find the surface area of a solid of revolution using polar coordinates?
  
• How do I find the surface area of the solid defined by revolving #r = 3sin(theta)# about the polar axis?
  
• How do I determine the volume of the solid obtained by revolving the curve #r=3sin(theta)# around the polar axis?
  
• How do I determine if the alternating series #sum_(n=1)^oo(-1)^n/sqrt(3n+1)# is convergent?
  
• How do you know when to use the Root Test for convergence of a series?
  
• When testing for convergence, how do you determine which test to use?
  
• What is the radius of convergence?
  
• How do you find the antiderivative of a power series?
  
• What is a Taylor series?
  
• What are the most important power series to memorise?
  
• How do I use a power series to calculate a limit?
  
• How do I find #lim_(x->oo)(3sin(x))/e^x# using power series?
  
• What uses do products of power series have?
  
• What is the formula for binomial expansion?
  
• What is the link between binomial expansions and Pascal's Triangle?
  
• If I want to test the series #sum_(n=1)^oo(n^2+2^n)/(1-e^(n+1))# for convergence, what would be the best test to use and why?
  
• When using integration to find an area, exactly which "area" is found?
  
• Is there a difference between #lim_(h->0)(f(x+h)-f(x))/h# and #lim_(deltax->0)(f(x+deltax)-f(x))/(deltax)#?
  
• What is the limit as x approaches 0 of #sin^2(x/x)#?
  
• What is the derivative of #x=y^2#?
  
• What is the derivative of #e^(9x)#?
  
• What is the limit as x approaches 0 from the right-hand side of #ln(x)#?
  
• What is the limit as x approaches infinity of a constant?
  
• What is the derivative of #|x|#?
  
• What is the limit as x approaches infinity of #6cos(x)#?
  
• What is the integral of #ln(x^2)#?
  
• What is the derivative of #e^3#?
  
• What is the integral of a constant?
  
• What is the antiderivative of the distance function?
  
• What is the derivative of #y=6xy#?
  
• What is the integral of #ln(7x)#?
  
• What is the limit as x approaches infinity of #1.001^x#?
  
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