Calculus Team Rapid Jeopardy

Applications of Derivatives

100

It is the derivative of the function at the given value

y = x² + 4x at x = −5  

y = x² + 4x at x = −5 

 dx/dy x=-5   =-6

       What is -6

200

It is the slope of the function at the given value.

y = x² + 6x + 7 at x = −2

y = x² + 6x + 7

y'=2x+6

y'=2(-2) + 6

What is 2

300

It is the equation of the line tangent to the function at the given point. Your answer should be in slope-intercept form

y = x³ − 3x² + 2 at (3, 2)

y = x³ − 3x² + 2

y'=3x²- 6x

y'=3(3)²-6(3) =9   y=9x+b  

(2)=9(3)+b 

2=27+b   b=-25     What is y=9x-25

400

It is the values of c that satisfy the Mean Value Theorem

y = −x² + 8x − 17 ; [3, 6] 

y = −x² + 8x − 17

y'= -2x + 8

-1 = -2c + 8

What is 9/2

500

It is the slope of the function at the given value.

y = −ln (−x + 2) at x = −3

y = −ln (−x + 2)

y'= 1/-x+2

y'= 1/-(-3)+2

What is 1/5

Antiderivatives

100

It is the Anti Derivative of:

3x²

What is x³ + c

200

It is the Anti Derivative of: 

sec²(x)

What is tan(x) + c

300

It is the Anti Derivative of: 

1/(1-x²)^1/2

What is sin^-1(x) + c

400

It is the Anti Derivative of: 

sec(x)

What is ln|sec(x)+tan(x)| + c

500

It is the Anti Derivative of: 

1/(a²+x²)

What is 1/a tan^-1(x/a) + c

Definite Integrals

100

It is the evaluation of the definite integral.

y = − x²/2 + x − 1/2 ; [−2, 1]

What is -1/2

200

It is the evaluation of the definite integral.

y = x²/2 − 2x − 1; [−1, 1]

What is 0

300

It is the evaluation of the definite integral.

y = x³ + 3x² − 2; [−2, 0]

What is { (−3 + (3)^1/2)/3 , (−3 − (3)^1/2)/3 } 

400

It is the evaluation of the definite integral.

y = −x³ + 4x² − 3; [0, 4]

What is 8/3

500

It is the evaluation of the definite integral.

y = −(−5x + 25)^1/2 ; [3, 5]

What is 9/2

Area Under a Curve

100

It is the area under the curve over the given interval.

y=4/x² ; [-2,-1]

I^-1_-2 4/x² dx

What is 2

200

It is the area under the curve over the given interval.

y=sec²(x) ; [-pi , -3pi/4]

I^-pi_-3pi/4 sec²(x) dx

What is 1

300

It is the area under the curve over the given interval.

y=(x)^1/2

I⁵₄ (x)^1/2 dx

What is 2(5(5)^1/2-8) / 3 = 2.12

400

It is the area under the curve over the given interval.

y=-cos(x) ; [3pi/4 , pi]

I^pi_3pi/4 -cos(x) dx

(2)^1/2/2

What is 0.707

500

It is the area under the curve over the given interval.

y=6x²⁺6x ; [-.5 , .5]

What is 0

U Substitutions

100

It is the U value of the substitution.

I²₀ 2x[x²+4]²dx  

What is u=x²+4

200

It is the U value of the substitution.

I⁰_-1 18x²(3x²+3)² dx 

What is u=3x²+3

300

It is the derivative of the U value of the substitution.

I²₀ 2x[x²+4]²dx  

U=x²+4

du=2x dx

What is du= 2x dx

400

It is the integral of this equation.

I¹₀ 16x/(4x²+4)² dx ; u=4x²+4

What is 1/4=.25

500

It is the integral of this expression.

I (3x+2)⁵ dx

(3x+2)⁶/18 +c