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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