Derivatives and integration

Question 1

If the variable cannot be eliminated in a parametric equation, how do you find dy/dx?

Answer 1

dy/dx = dy/dt / dx/dt

Question 2

For parametric equations, d²y / dx² =

Answer 2

d²y / dx² = d/dt(dy/dx) / dx/dt

Question 3

d/dx tanx = ?

Answer 3

d/dx tanx = sec²x

Question 4

d/dx secx = ?

Answer 4

d/dx secx = secxtanx

Question 5

d/dx arsinx =

d/dx arcosx =

Answer 5

d/dx arsinx = 1 / √(1 - x²)

d/dx arcosx = - 1 / √(1 - x²)

dasp - d/dx arsin is positive

dacn - d/dx arsin is negative

Question 6

d/dx a^x =

Answer 6

d/dx a^x = a^xln(a)

Question 7

d/dx ln(x) =

Answer 7

d/dx ln(x) = 1/x

Question 8

d/dx tanhx =

Answer 8

d/dx tanhx = sech²x

Question 9

What is the formula for integration by parts?

Answer 9

∫udv = uv - ∫vdu

Always let u be the function that simplifies when differentiated.

Question 10

When is integration by substitution used?

Answer 10

To integrate a composite function.

Subsitute one function for f(x) = u and attempt to get a standard integral form. Where possible, let u = something whose derivative is on the top line (so it cancels).

Question 11

What is the difference between odd and even powered trigonometric integrals?

Answer 11

Even powers – use double angle formulae and identities.

Odd powers – transform into a single trig function.

Question 12

Trigonometric substitutions.

√(a² - x²) use ?

√(a² + x²) use ?

√(x² - a²) use ?

Answer 12

√(a² - x²) use x = asinø results in acosø

√(a² + x²) use x = atanø results in asecø

√(x2 - a2) use x = asecø results in atanø