Practice exam · Calculus I · Units 1–3A

Calculus I Practice Final Exam with Complete Solutions

A balanced twenty-five-question cumulative final with point values, study map, and complete solutions.

25 problems120 min estimated timeCumulative progression

What is included

Prepare for a Calculus I final with a timed BetterGrades practice exam covering limits through integral foundations.

Skills assessed

course synthesis
method selection
mathematical communication

Prerequisites

Calculus I Units 1 through 3A

Exam conditions

Suggested time: 120 minutes

Points: 4 points per question; 100 points total

Calculator: Scientific calculator permitted; computer algebra is not required.

This is an original BetterGrades practice exam, not a released institutional exam.

Printable and accessible

Download this resource

No email address or account is required.

Student PDF Worked answer key PDF

Device-only study control

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Evaluate $\lim_{x\to2}(x^2+3x-1)$ .
Evaluate $\lim_{x\to4}(x^2-16)/(x-4)$ .
Choose k so $f(x)=kx+1$ for x<2 and $f(x)=7$ for x≥2 is continuous.
Use the limit definition to find the derivative of x².
Differentiate $4x^5-3x^2+7$ .
Differentiate $x^2\sin x$ .
Differentiate $(x+1)/(x-1)$ .
Differentiate $(3x^2+1)^4$ .
For x²+y²=25, find dy/dx.
Find the tangent line to y=x³ at x=2.
Find c guaranteed by MVT for f(x)=x² on [1,3].
Find critical numbers of x³-3x.
Where is x³-3x increasing?
Find the inflection point of x³-6x².
A rectangle has perimeter 40. Find maximum area.
A circle radius grows at 2 cm/s. Find dA/dt at r=5.
Use linearization at 9 to approximate √9.2.
Find $\int(6x^2-4)dx$ .
Evaluate $\int_0^2 3x^2dx$ .
Differentiate $F(x)=\int_1^x\cos(t^2)dt$ .
Evaluate $\int2x(x^2+1)^3dx$ .
Find the area under y=x on [0,3].
Find the average value of x² on [0,3].
If v(t)=3t²-6t, find displacement from 0 to 3.
Explain why a differentiable function is continuous.

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Evaluate $\lim_{x\to2}(x^2+3x-1)$ .

Answer: $9$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Direct substitution applies.
The verified result is $9$ .

Problem 2: Evaluate $\lim_{x\to4}(x^2-16)/(x-4)$ .

Answer: $8$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Factor and cancel x-4.
The verified result is $8$ .

Problem 3: Choose k so $f(x)=kx+1$ for x<2 and $f(x)=7$ for x≥2 is continuous.

Answer: $k=3$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Match the left limit 2k+1 to 7.
The verified result is $k=3$ .

Problem 4: Use the limit definition to find the derivative of x².

Answer: $2x$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Expand (x+h)², cancel, divide by h, and let h→0.
The verified result is $2x$ .

Problem 5: Differentiate $4x^5-3x^2+7$ .

Answer: $20x^4-6x$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Apply linearity and the power rule.
The verified result is $20x^4-6x$ .

Problem 6: Differentiate $x^2\sin x$ .

Answer: $2x\sin x+x^2\cos x$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use the product rule.
The verified result is $2x\sin x+x^2\cos x$ .

Problem 7: Differentiate $(x+1)/(x-1)$ .

Answer: $-2/(x-1)^2$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use the quotient rule and simplify.
The verified result is $-2/(x-1)^2$ .

Problem 8: Differentiate $(3x^2+1)^4$ .

Answer: $24x(3x^2+1)^3$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Outer power times inner derivative.
The verified result is $24x(3x^2+1)^3$ .

Problem 9: For x²+y²=25, find dy/dx.

Answer: $-x/y$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Differentiate both sides with respect to x.
The verified result is $-x/y$ .

Problem 10: Find the tangent line to y=x³ at x=2.

Answer: $y-8=12(x-2)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Evaluate the point and derivative.
The verified result is $y-8=12(x-2)$ .

Problem 11: Find c guaranteed by MVT for f(x)=x² on [1,3].

Answer: $c=2$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Set 2c equal to the secant slope 4.
The verified result is $c=2$ .

Problem 12: Find critical numbers of x³-3x.

Answer: $x=\pm1$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Solve 3x²-3=0.
The verified result is $x=\pm1$ .

Problem 13: Where is x³-3x increasing?

Answer: $(-\infty,-1)\cup(1,\infty)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use the sign of 3(x²-1).
The verified result is $(-\infty,-1)\cup(1,\infty)$ .

Problem 14: Find the inflection point of x³-6x².

Answer: $(2,-16)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

f''=6x-12 changes sign at 2.
The verified result is $(2,-16)$ .

Problem 15: A rectangle has perimeter 40. Find maximum area.

Answer: $100$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

A=x(20-x) is maximal at x=10.
The verified result is $100$ .

Problem 17: Use linearization at 9 to approximate √9.2.

Answer: $3+0.2/6\approx3.0333$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

L(x)=3+(x-9)/6.
The verified result is $3+0.2/6\approx3.0333$ .

Problem 18: Find $\int(6x^2-4)dx$ .

Answer: $2x^3-4x+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Integrate term by term.
The verified result is $2x^3-4x+C$ .

Problem 19: Evaluate $\int_0^2 3x^2dx$ .

Answer: $8$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use x³ at the endpoints.
The verified result is $8$ .

Problem 20: Differentiate $F(x)=\int_1^x\cos(t^2)dt$ .

Answer: $\cos(x^2)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Apply FTC Part I.
The verified result is $\cos(x^2)$ .

Problem 21: Evaluate $\int2x(x^2+1)^3dx$ .

Answer: $(x^2+1)^4/4+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Let u=x²+1.
The verified result is $(x^2+1)^4/4+C$ .

Problem 22: Find the area under y=x on [0,3].

Answer: $9/2$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Integrate x or use triangle area.
The verified result is $9/2$ .

Problem 23: Find the average value of x² on [0,3].

Answer: $3$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Divide the integral 9 by interval length 3.
The verified result is $3$ .

Problem 24: If v(t)=3t²-6t, find displacement from 0 to 3.

Answer: $0$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Integrate velocity over the interval.
The verified result is $0$ .

Problem 25: Explain why a differentiable function is continuous.

Answer: $The derivative limit forces f(x)-f(a) to approach 0.$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Factor f(x)-f(a) as a difference quotient times x-a.
The verified result is $The derivative limit forces f(x)-f(a) to approach 0.$ .

Common errors

Starting with a formula before identifying the structure.
Skipping hypotheses or endpoint checks.
Giving an answer without enough reasoning to audit it.

Practice exam · Calculus I · Units 1–3A

Calculus I Practice Final Exam with Complete Solutions

A balanced twenty-five-question cumulative final with point values, study map, and complete solutions.

25 problems120 min estimated timeCumulative progression

What is included

Prepare for a Calculus I final with a timed BetterGrades practice exam covering limits through integral foundations.

Skills assessed

course synthesis
method selection
mathematical communication

Prerequisites

Calculus I Units 1 through 3A

Exam conditions

Suggested time: 120 minutes

Points: 4 points per question; 100 points total

Calculator: Scientific calculator permitted; computer algebra is not required.

This is an original BetterGrades practice exam, not a released institutional exam.

Printable and accessible

Download this resource

No email address or account is required.

Student PDF Worked answer key PDF

Device-only study control

Ready for an honest first attempt?

This control sends only the resource id and type—never your answers or identity.

Printable preview

Evaluate $\lim_{x\to2}(x^2+3x-1)$ .
Evaluate $\lim_{x\to4}(x^2-16)/(x-4)$ .
Choose k so $f(x)=kx+1$ for x<2 and $f(x)=7$ for x≥2 is continuous.
Use the limit definition to find the derivative of x².
Differentiate $4x^5-3x^2+7$ .
Differentiate $x^2\sin x$ .
Differentiate $(x+1)/(x-1)$ .
Differentiate $(3x^2+1)^4$ .
For x²+y²=25, find dy/dx.
Find the tangent line to y=x³ at x=2.
Find c guaranteed by MVT for f(x)=x² on [1,3].
Find critical numbers of x³-3x.
Where is x³-3x increasing?
Find the inflection point of x³-6x².
A rectangle has perimeter 40. Find maximum area.
A circle radius grows at 2 cm/s. Find dA/dt at r=5.
Use linearization at 9 to approximate √9.2.
Find $\int(6x^2-4)dx$ .
Evaluate $\int_0^2 3x^2dx$ .
Differentiate $F(x)=\int_1^x\cos(t^2)dt$ .
Evaluate $\int2x(x^2+1)^3dx$ .
Find the area under y=x on [0,3].
Find the average value of x² on [0,3].
If v(t)=3t²-6t, find displacement from 0 to 3.
Explain why a differentiable function is continuous.

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Evaluate $\lim_{x\to2}(x^2+3x-1)$ .

Answer: $9$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Direct substitution applies.
The verified result is $9$ .

Problem 2: Evaluate $\lim_{x\to4}(x^2-16)/(x-4)$ .

Answer: $8$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Factor and cancel x-4.
The verified result is $8$ .

Problem 3: Choose k so $f(x)=kx+1$ for x<2 and $f(x)=7$ for x≥2 is continuous.

Answer: $k=3$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Match the left limit 2k+1 to 7.
The verified result is $k=3$ .

Problem 4: Use the limit definition to find the derivative of x².

Answer: $2x$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Expand (x+h)², cancel, divide by h, and let h→0.
The verified result is $2x$ .

Problem 5: Differentiate $4x^5-3x^2+7$ .

Answer: $20x^4-6x$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Apply linearity and the power rule.
The verified result is $20x^4-6x$ .

Problem 6: Differentiate $x^2\sin x$ .

Answer: $2x\sin x+x^2\cos x$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use the product rule.
The verified result is $2x\sin x+x^2\cos x$ .

Problem 7: Differentiate $(x+1)/(x-1)$ .

Answer: $-2/(x-1)^2$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use the quotient rule and simplify.
The verified result is $-2/(x-1)^2$ .

Problem 8: Differentiate $(3x^2+1)^4$ .

Answer: $24x(3x^2+1)^3$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Outer power times inner derivative.
The verified result is $24x(3x^2+1)^3$ .

Problem 9: For x²+y²=25, find dy/dx.

Answer: $-x/y$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Differentiate both sides with respect to x.
The verified result is $-x/y$ .

Problem 10: Find the tangent line to y=x³ at x=2.

Answer: $y-8=12(x-2)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Evaluate the point and derivative.
The verified result is $y-8=12(x-2)$ .

Problem 11: Find c guaranteed by MVT for f(x)=x² on [1,3].

Answer: $c=2$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Set 2c equal to the secant slope 4.
The verified result is $c=2$ .

Problem 12: Find critical numbers of x³-3x.

Answer: $x=\pm1$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Solve 3x²-3=0.
The verified result is $x=\pm1$ .

Problem 13: Where is x³-3x increasing?

Answer: $(-\infty,-1)\cup(1,\infty)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use the sign of 3(x²-1).
The verified result is $(-\infty,-1)\cup(1,\infty)$ .

Problem 14: Find the inflection point of x³-6x².

Answer: $(2,-16)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

f''=6x-12 changes sign at 2.
The verified result is $(2,-16)$ .

Problem 15: A rectangle has perimeter 40. Find maximum area.

Answer: $100$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

A=x(20-x) is maximal at x=10.
The verified result is $100$ .

Problem 17: Use linearization at 9 to approximate √9.2.

Answer: $3+0.2/6\approx3.0333$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

L(x)=3+(x-9)/6.
The verified result is $3+0.2/6\approx3.0333$ .

Problem 18: Find $\int(6x^2-4)dx$ .

Answer: $2x^3-4x+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Integrate term by term.
The verified result is $2x^3-4x+C$ .

Problem 19: Evaluate $\int_0^2 3x^2dx$ .

Answer: $8$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Use x³ at the endpoints.
The verified result is $8$ .

Problem 20: Differentiate $F(x)=\int_1^x\cos(t^2)dt$ .

Answer: $\cos(x^2)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Apply FTC Part I.
The verified result is $\cos(x^2)$ .

Problem 21: Evaluate $\int2x(x^2+1)^3dx$ .

Answer: $(x^2+1)^4/4+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Let u=x²+1.
The verified result is $(x^2+1)^4/4+C$ .

Problem 22: Find the area under y=x on [0,3].

Answer: $9/2$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Integrate x or use triangle area.
The verified result is $9/2$ .

Problem 23: Find the average value of x² on [0,3].

Answer: $3$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Divide the integral 9 by interval length 3.
The verified result is $3$ .

Problem 24: If v(t)=3t²-6t, find displacement from 0 to 3.

Answer: $0$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Integrate velocity over the interval.
The verified result is $0$ .

Problem 25: Explain why a differentiable function is continuous.

Answer: $The derivative limit forces f(x)-f(a) to approach 0.$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Factor f(x)-f(a) as a difference quotient times x-a.
The verified result is $The derivative limit forces f(x)-f(a) to approach 0.$ .

Common errors

Starting with a formula before identifying the structure.
Skipping hypotheses or endpoint checks.
Giving an answer without enough reasoning to audit it.

What is included

Skills assessed

Prerequisites

Exam conditions

Download this resource

Ready for an honest first attempt?

Printable preview

Complete worked solutions

Problem 1: Evaluate lim⁡x→2(x2+3x−1)\lim_{x\to2}(x^2+3x-1)limx→2​(x2+3x−1).

Problem 2: Evaluate lim⁡x→4(x2−16)/(x−4)\lim_{x\to4}(x^2-16)/(x-4)limx→4​(x2−16)/(x−4).

Problem 3: Choose k so f(x)=kx+1f(x)=kx+1f(x)=kx+1 for x<2 and f(x)=7f(x)=7f(x)=7 for x≥2 is continuous.

Problem 4: Use the limit definition to find the derivative of x².

Problem 5: Differentiate 4x5−3x2+74x^5-3x^2+74x5−3x2+7.

Problem 6: Differentiate x2sin⁡xx^2\sin xx2sinx.

Problem 7: Differentiate (x+1)/(x−1)(x+1)/(x-1)(x+1)/(x−1).

Problem 8: Differentiate (3x2+1)4(3x^2+1)^4(3x2+1)4.

Problem 9: For x²+y²=25, find dy/dx.

Problem 10: Find the tangent line to y=x³ at x=2.

Problem 11: Find c guaranteed by MVT for f(x)=x² on [1,3].

Problem 12: Find critical numbers of x³-3x.

Problem 13: Where is x³-3x increasing?

Problem 14: Find the inflection point of x³-6x².

Problem 15: A rectangle has perimeter 40. Find maximum area.

Problem 16: A circle radius grows at 2 cm/s. Find dA/dt at r=5.

Problem 17: Use linearization at 9 to approximate √9.2.

Problem 18: Find ∫(6x2−4)dx\int(6x^2-4)dx∫(6x2−4)dx.

Problem 19: Evaluate ∫023x2dx\int_0^2 3x^2dx∫02​3x2dx.

Problem 20: Differentiate F(x)=∫1xcos⁡(t2)dtF(x)=\int_1^x\cos(t^2)dtF(x)=∫1x​cos(t2)dt.

Problem 21: Evaluate ∫2x(x2+1)3dx\int2x(x^2+1)^3dx∫2x(x2+1)3dx.

Problem 22: Find the area under y=x on [0,3].

Problem 23: Find the average value of x² on [0,3].

Problem 24: If v(t)=3t²-6t, find displacement from 0 to 3.

Problem 25: Explain why a differentiable function is continuous.

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

What is included

Skills assessed

Prerequisites

Exam conditions

Download this resource

Ready for an honest first attempt?

Printable preview

Complete worked solutions

Problem 1: Evaluate lim⁡x→2(x2+3x−1)\lim_{x\to2}(x^2+3x-1)limx→2​(x2+3x−1).

Problem 2: Evaluate lim⁡x→4(x2−16)/(x−4)\lim_{x\to4}(x^2-16)/(x-4)limx→4​(x2−16)/(x−4).

Problem 3: Choose k so f(x)=kx+1f(x)=kx+1f(x)=kx+1 for x<2 and f(x)=7f(x)=7f(x)=7 for x≥2 is continuous.

Problem 4: Use the limit definition to find the derivative of x².

Problem 5: Differentiate 4x5−3x2+74x^5-3x^2+74x5−3x2+7.

Problem 6: Differentiate x2sin⁡xx^2\sin xx2sinx.

Problem 7: Differentiate (x+1)/(x−1)(x+1)/(x-1)(x+1)/(x−1).

Problem 8: Differentiate (3x2+1)4(3x^2+1)^4(3x2+1)4.

Problem 9: For x²+y²=25, find dy/dx.

Problem 10: Find the tangent line to y=x³ at x=2.

Problem 11: Find c guaranteed by MVT for f(x)=x² on [1,3].

Problem 12: Find critical numbers of x³-3x.

Problem 13: Where is x³-3x increasing?

Problem 14: Find the inflection point of x³-6x².

Problem 15: A rectangle has perimeter 40. Find maximum area.

Problem 16: A circle radius grows at 2 cm/s. Find dA/dt at r=5.

Problem 17: Use linearization at 9 to approximate √9.2.

Problem 18: Find ∫(6x2−4)dx\int(6x^2-4)dx∫(6x2−4)dx.

Problem 19: Evaluate ∫023x2dx\int_0^2 3x^2dx∫02​3x2dx.

Problem 20: Differentiate F(x)=∫1xcos⁡(t2)dtF(x)=\int_1^x\cos(t^2)dtF(x)=∫1x​cos(t2)dt.

Problem 21: Evaluate ∫2x(x2+1)3dx\int2x(x^2+1)^3dx∫2x(x2+1)3dx.

Problem 22: Find the area under y=x on [0,3].

Problem 23: Find the average value of x² on [0,3].

Problem 24: If v(t)=3t²-6t, find displacement from 0 to 3.

Problem 25: Explain why a differentiable function is continuous.

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

Problem 1: Evaluate $\lim_{x\to2}(x^2+3x-1)$ .

Problem 2: Evaluate $\lim_{x\to4}(x^2-16)/(x-4)$ .

Problem 3: Choose k so $f(x)=kx+1$ for x<2 and $f(x)=7$ for x≥2 is continuous.

Problem 5: Differentiate $4x^5-3x^2+7$ .

Problem 6: Differentiate $x^2\sin x$ .

Problem 7: Differentiate $(x+1)/(x-1)$ .

Problem 8: Differentiate $(3x^2+1)^4$ .

Problem 18: Find $\int(6x^2-4)dx$ .

Problem 19: Evaluate $\int_0^2 3x^2dx$ .

Problem 20: Differentiate $F(x)=\int_1^x\cos(t^2)dt$ .

Problem 21: Evaluate $\int2x(x^2+1)^3dx$ .

Problem 1: Evaluate $\lim_{x\to2}(x^2+3x-1)$ .

Problem 2: Evaluate $\lim_{x\to4}(x^2-16)/(x-4)$ .

Problem 3: Choose k so $f(x)=kx+1$ for x<2 and $f(x)=7$ for x≥2 is continuous.

Problem 5: Differentiate $4x^5-3x^2+7$ .

Problem 6: Differentiate $x^2\sin x$ .

Problem 7: Differentiate $(x+1)/(x-1)$ .

Problem 8: Differentiate $(3x^2+1)^4$ .

Problem 18: Find $\int(6x^2-4)dx$ .

Problem 19: Evaluate $\int_0^2 3x^2dx$ .

Problem 20: Differentiate $F(x)=\int_1^x\cos(t^2)dt$ .

Problem 21: Evaluate $\int2x(x^2+1)^3dx$ .