AP Calc BC Chapter 6 Practice Problems

find the area of the region bounded by the coordinate axes, the line y=3, and the graph of x=y²+2 with respect to y

Chap. 5 Test, 1 determine where the curves intersect to find bounds determine which curve is "on top" plug into area EQ and solve

find the volume of the shape generated by revolving the area between the curve y= 1/x₂ and the x-axis about the x-axis

Chap. 5 Test, 3 determine if/where curves intersect to find bounds determine which curve is "on top" plug into volume EQ and solve

find y': y=x² * log₂(3-2x)

HW #7, 15 logarithmic derivative rule and chain rule

condense: 2ln(x+1) + 1/3 * lnx - ln(cosx)

HW #7, 15 take coefficients and move to power spot then make into 1 log (pos. on top, neg. on bottom)

find the derivative: y = x∧sinx

HW #7, 32 take the ln of both sides, derive implicitly

integrate: ∫[2/x + 3sinx] dx

HW #7, 59 logarithmic integration rule and chain rule

integrate: ∫(t+1)/t dt

HW #7, 65 split up fraction and integrate terms separately with logarithmic integration rules

integrate: ∫e∧2x dx

HW #7, 65 u substitution

integrate: ∫²₀ 3x/1+x² dx

HW #7, 67 (the first) u substitution and logarithmic integration rule

integrate: ∫e∧sinx * cosx dx

HW #7, 67 (the second) u substitution

Sketch the curve y=1/t and shade a region under the curve whose area is a) ln 2 b) -ln0.5 c) 2.

HW #8, 1 write integral of curve to find area beneath determine bounds, keeping in mind that ln1=0 a) bounds: 1-2 b) bounds: 0.5-1 (1 on top gets negative number) c) bounds: 1-e² (lne² becomes 2lne which is just 2)

find dy/dx: y= tan⁻¹(x³)

HW #8, 19 plug into derivative of inverse trig function equation and chain rule

find dy/dx: y=sec⁻¹(x⁵)

HW #8, 20 plug into derivative of inverse trig function equation and chain rule

find the integral: ∫[1/2√1-x² - 3/1+x²] dx

HW #8, 29 split up into separate integrals, then use the derivative of inverse trig function equations

find the limit: lim(x→infinity) [cos(2/x)]∧x²

HW #8, 30 (the first) plug in 0, put into a good indeterminate form, then use L'Hospitals rule (twice)

evaluate the integral: ∫1/√1-4x² dx

HW #8, 31 take out the square from 4x² so that it is (2x)² use u substitution where u = 2x use derivative of inverse trig function equation to integrate

evaluate the integral: ∫sinx/(cos²x+1) dx

HW #8, 36 rearrange denominated so square is on the right u substitution where u=cosx use derivative of inverse trig function equation to integrate

evaluate the integral: ∫√₂ ² 1/x√x²-1 dx

HW #8, 39 use derivative of inverse trig function equation to integrate plug in bounds (top-bottom) set up triangle so that sec⁻¹x=2 and √2 to find angles plug in angles

find the exact value: sec[sin⁻¹(-3/4)]

HW #8, 4 (the second) draw a triangle where sinx=3/4, fill in the third side with Pythagorean theorem, then find secant of that triangle

evaluate the integral: ∫₁ ∧√e 1/x√1-(lnx)² dx

HW #8, 42 u substitution where u=lnx use derivative of inverse trig function equation to integrate plug in bounds (top-bottom) set up triangle so that sec⁻¹x=2 and √2 to find angles plug in angles

find the exact value: sin[2cos⁻¹(3/5)]

HW #8, 5 call stuff in brackets "theta" and set up double sine formula, draw a triangle where cos=3/5 and fill in the third side with Pythagorean theorem, then find sine and cosine of that triangle and plug those into the double angle formula

find the limit: lim(x→0) (e∧x-1)/sinx

HW #8, 7 plug in 0 then use L'Hospitals rule

find y': y=ln(cos(e∧x))

Quiz, 1 logarithmic derivative rule and chain rule

find y': y = e∧³√2=5x²

Quiz, 2 logarithmic derivative rule and chain rule