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.

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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.

01

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.

The polynomial is continuous, so substitution gives $2^2+3(2)-1=4+6-1=9$ .
The verified result is $9$ .

02

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.

For $x\ne4$ , $(x^2-16)/(x-4)=x+4$ ; therefore the limit is $4+4=8$ .
The verified result is $8$ .

03

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.

Continuity requires $2k+1=7$ , so $2k=6$ and $k=3$ .
The verified result is $k=3$ .

04

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.

$f'(x)=\lim_{h\to0}((x+h)^2-x^2)/h=\lim_{h\to0}(2x+h)=2x$ .
The verified result is $2x$ .

05

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.

$d(4x^5)/dx=20x^4$ , $d(-3x^2)/dx=-6x$ , and the constant derivative is zero.
The verified result is $20x^4-6x$ .

06

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.

The product rule gives $(x^2)'\sin x+x^2(\sin x)'=2x\sin x+x^2\cos x$ .
The verified result is $2x\sin x+x^2\cos x$ .

07

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.

The quotient rule gives $((x-1)-(x+1))/(x-1)^2=-2/(x-1)^2$ , for $x\ne1$ .
The verified result is $-2/(x-1)^2$ .

08

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.

With $u=3x^2+1$ , $d(u^4)/dx=4u^3(6x)=24x(3x^2+1)^3$ .
The verified result is $24x(3x^2+1)^3$ .

09

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.

Differentiating gives $2x+2y\,dy/dx=0$ , hence $dy/dx=-x/y$ where $y\ne0$ .
The verified result is $-x/y$ .

10

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.

At $x=2$ , $y=8$ and $y'=3x^2=12$ , so point-slope form is $y-8=12(x-2)$ .
The verified result is $y-8=12(x-2)$ .

11

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.

The secant slope is $(9-1)/(3-1)=4$ ; solving $f'(c)=2c=4$ gives $c=2\in(1,3)$ .
The verified result is $c=2$ .

12

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

Answer: $x=\pm1$

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

$f'(x)=3x^2-3=3(x-1)(x+1)$ , so the critical numbers are $x=-1,1$ .
The verified result is $x=\pm1$ .

13

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.

Since $f'(x)=3(x^2-1)>0$ exactly when $|x|>1$ , the function increases on $( -\infty,-1)\cup(1,\infty)$ .
The verified result is $(-\infty,-1)\cup(1,\infty)$ .

14

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.

$f''(x)=6x-12$ changes from negative to positive at $x=2$ , and $f(2)=8-24=-16$ .
The verified result is $(2,-16)$ .

15

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.

If the sides are $x$ and $20-x$ , then $A=x(20-x)$ ; $A'=20-2x=0$ at $x=10$ , giving $A=100$ .
The verified result is $100$ .

17

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.

For $f(x)=\sqrt{x}$ , $f(9)=3$ and $f'(9)=1/6$ , so $L(9.2)=3+0.2/6\approx3.0333$ .
The verified result is $3+0.2/6\approx3.0333$ .

18

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.

Termwise integration gives $6x^3/3-4x+C=2x^3-4x+C$ .
The verified result is $2x^3-4x+C$ .

19

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

Answer: $8$

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

An antiderivative is $x^3$ , so $[x^3]_0^2=8-0=8$ .
The verified result is $8$ .

20

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.

FTC Part I evaluates the integrand at the variable upper bound: $F'(x)=\cos(x^2)$ .
The verified result is $\cos(x^2)$ .

21

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 $u=x^2+1$ , so $du=2x\,dx$ ; then $\int u^3du=u^4/4+C=(x^2+1)^4/4+C$ .
The verified result is $(x^2+1)^4/4+C$ .

22

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.

$A=\int_0^3x\,dx=[x^2/2]_0^3=9/2$ .
The verified result is $9/2$ .

23

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.

$f_{\rm avg}=\frac1{3-0}\int_0^3x^2dx=\frac13[x^3/3]_0^3=3$ .
The verified result is $3$ .

24

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.

$\int_0^3(3t^2-6t)dt=[t^3-3t^2]_0^3=(27-27)-0=0$ .
The verified result is $0$ .

25

Problem 25: Explain why a differentiable function is continuous.

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

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

Write $f(x)-f(a)=((f(x)-f(a))/(x-a))(x-a)$ ; differentiability makes the first factor approach $f'(a)$ while the second approaches zero.
The verified result is $\text{The derivative limit forces }f(x)-f(a)\text{ to approach }0.$ .

Common errors

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