DIFF. CALCULUS PRACTICE TEST

Determine the domain of f(x)=ln(4 − x²).

(-2, 2)

Find the domain of f(x)=√(x−2)/(x²−9).

[2,3) ∪ (3,∞)

State the range of f(x)=3x²−12.

[-12, ∞)

Evaluate lim(x→2) (x²−4)/(x−2).

4

Evaluate lim(x→0) sin(5x)/x.

5

Evaluate lim(x→∞) (3x²−x+1)/(5x²+7).

3/5

Evaluate lim(x→1⁻) floor(3x).

2

Determine if lim(x→0) |x|/x exists.

DNE

Find a and b so that f(x) is continuous at x=3 for f(x)=2x−1 (x<3), ax+b (x≥3).

3a + b = 5

Identify the type of discontinuity in f(x)=(x²−9)/(x−3).

Removable discontinuity (hole)

Determine whether f(x)=|x| is continuous at x=0.

Yes

Find dy/dx of y=sin(3x²).

6x cos(3x²)

Differentiate y=x^x.

x^x(1 + ln x)

Find dy/dx of y=ln(5x²+3).

10x/(5x²+3)

Find dy/dx of y=(2x+3)/(x−4).

-11/(x−4)²

Differentiate y=e^(2x)cos(x).

e^{2x}(2 cos x − sin x)

Find dy/dx for x² + xy − y² = 10.

(-2x - y)/(x - 2y)

Differentiate implicitly: x³ + y³ = 6xy.

(6y - 3x²)/(3y² - 6x) = (2y - x²)/(y² - 2x)

Find the tangent line of y=4x²−x+1 at x=2.

y = 15x - 15

If f′(x)>0 on (1,5), what does this imply about f(x)?

f is increasing on (1,5)

Find the critical points of f(x)=x³ − 6x² + 9x.

x = 1, 3

Find where f(x)=x⁴ − 2x² is increasing.

(-1,0) ∪ (1, ∞)

A rectangle has perimeter 40 m. Find dimensions that maximize area.

10 m × 10 m (square)

Find the maximum area of a right triangle with hypotenuse 10.

25

A farmer has 100 m of fencing. Find dimensions maximizing area.

25 m × 25 m (square)

A balloon is inflated at 20 cm³/s. Find dr/dt when r=5 cm.

1/(5π) cm/s

A 10 m ladder slides; bottom moves 2 m/s. Find dy/dt when bottom is 6 m away.

-1.5 m/s

Water depth decreases at 3 cm/s; find volume rate change when depth=40 cm.

dV/dt = A · dh/dt = A(−3) ⇒ dV/dt = −3A (A = cross-sectional area)

Find f″(x) for f(x)=(x²+1)³.

6(x²+1)(5x²+1)

Find inflection points of f(x)=x⁴−4x².

x = ±√(2/3)

If f″(x)<0 on an interval, what does this mean?

f is concave down on that interval

Find dy/dx for x=3t²−t, y=2t³.

(6t²)/(6t − 1)

Find slope at t=1 for x=sin t, y=cos t.

• tan(1)

Find speed of particle x=ln t, y=t² at t=2.

√65 / 2

Find ∂f/∂x for f=x²y + 4xy².

2xy + 4y²

Find ∂²f/∂x∂y for f=x³y + y².

3x²

Evaluate ∂f/∂y at (2,1) for f=xy³ + x²y.

10

State Clairaut’s theorem.

If second partials are continuous, ∂²f/∂x∂y = ∂²f/∂y∂x

Evaluate lim(x→0) (1−cos 4x)/x².

8

Find y''' for y=(3x−1)⁵.

1620(3x − 1)²

Find d²y/dx² for x=t³−t, y=t²+2.

-2(3t² + 1)/(3t² − 1)³

Determine where f(x)=ln(x²+4) is increasing.

x > 0

Find dy/dx of y=√(x²+3).

x / √(x² + 3)

Find where tangent to y=x³−3x is horizontal.

x = ±1

Find average rate of change of f(x)=x² + 3x + 2 from x=1 to x=4.

8