Calculus 1: Test 3 Review

Riemann Sums/Limit Def.

100

Find the area under the curve y = 2x-x² from x=1 to x=2 with n=4 left-endpoint rectangles.

area = 25/32

200

State the limit definition of a definite integral. 

Given on board.

300

Is a left-hand Riemann sum with 5 rectangles an under-approximation or an over-approximation for 2x+3 on the interval from 0 to 5?

Under-approximation

400

Using the limit definition of a definite integral, integrate with respect to x on the interval from 0 to 3: x³

81/4

Integrals

100

Integrate with respect to x: 2x+5

x²+5x+C

200

Integrate with respect to x: x^1/3(2+x)

(3x^4/3/2) + (3x^7/3/7) + C

300

Integrate with respect to x: (x²+1)(x³+3x)^-5

1/(12(x³+3x)⁴) + C

400

Integrate with respect to x: x-2sec²x

(x²/2)-2tanx+C

500

Integrate with respect to x: sin(sqrt(x))/sqrt(x)

-2cos(sqrt(x))+C

Derivatives

100

Find the derivative of: ln(x⁴+8)

4x³/(x⁴+8)

200

d/dx log₆(3xtanx)

(3tan(x)+3xsec²x)/(3xtanx)

300

Take the derivative of: e ^3x-ln(e^3x)

3e^3x-3

400

Take the derivative of: f(x) = 10^sinx

(10^sinx)(ln(10))(cosx)

500

Find the derivative: log₁₂(x³)

3/(x(ln(12)))

More Integrals

100

Integrate with respect to x: sin(sinx)cosx

-cos(sinx) + C

200

Integrate with respect to x: 1/x⁴

-1/3(x^-3)

300

Integrate with respect to x from 0 to 1: x⁴-5x³+3x²-4x-6

-161/20

400

Integrate with respect to x from pi/6 to 5(pi)/4: (sec²x)/tanx

Give the exact answer. 

ln(sqrt(3)/3)

500

Find the area under the curve on the interval from 1 to e: (1/x)(cos(lnx))

Give the exact answer (no decimals). 

sin(1)